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In how many ways can the letters of the word PERMUTATIONS be

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In how many ways can the letters of the word PERMUTATIONS be [#permalink]  17 May 2010, 13:01
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Here a couple of problems I encountered. Let me know your views on them. They may not necessarily be of GMAT format. Please share your views, nevertheless.

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Cheers
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Re: Interesting problems of Permutations and Combinations [#permalink]  18 May 2010, 00:13
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chandrun wrote:
Here a couple of problems I encountered. Let me know your views on them. They may not necessarily be of GMAT format. Please share your views, nevertheless.

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Cheers

THEORY:

Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:

$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$.

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.

Back to the original questions:

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

There are 12 letters in the word "PERMUTATIONS", out of which T is repeated twice.

1. Choosing 4 letters out of 10 (12-2(P and S)=10) to place between P and S = 10C4 = 210;
2. Permutation of the letters P ans S (PXXXXS or SXXXXP) = 2! =2;
3. Permutation of the 4 letters between P and S = 4! =24;
4. Permutations of the 7 units {P(S)XXXXS(P)}{X}{X}{X}{X}{X}{X} = 7! = 5040;
5. We should divide multiplication of the above 4 numbers by 2! as there is repeated T.

Hence: $$\frac{10C4*2!*4!*7!}{2!}=25,401,600$$

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

There are 11 letters in the word "MISSISSIPPI ", out of which: M=1, I=4, S=4, P=2.

Total # of permutations is $$\frac{11!}{4!4!2!}$$;
# of permutations with 4 I's together is $$\frac{8!}{4!2!}$$. Consider 4 I's as one unit: {M}{S}{S}{S}{S}{P}{P}{IIII} - total 8 units, out of which {M}=1, {S}=4, {P}=2, {IIII}=1.

So # of permutations with 4 I's not come together is: $$\frac{11!}{4!4!2!}-\frac{8!}{4!2!}$$.

Hope it helps.
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Re: Interesting problems of Permutations and Combinations [#permalink]  18 May 2010, 06:09
Thank you bunuel, Doing great job dude...
Hats off!!!!
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Re: Interesting problems of Permutations and Combinations [#permalink]  19 May 2010, 07:13
Thank you very much. That is an awesome piece of explanation.
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Re: Interesting problems of Permutations and Combinations [#permalink]  31 Jan 2011, 16:31
Why 7!? There are only six letters to arrange...
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Re: Interesting problems of Permutations and Combinations [#permalink]  31 Jan 2011, 16:45
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mariyea wrote:
Why 7!? There are only six letters to arrange...

Are you talking about Q1, 4th point? Can you please be more specific when asking questions?

4. Permutations of the 7 units {P(S)XXXXS(P)}{X}{X}{X}{X}{X}{X} = 7! = 5040;

There are 7 units: {PXXXXS}, {X}, {X}, {X}, {X}, {X}, {X} --> 6 letters in unit 1, plus 6 units with one letter in each = total of 12 letters.
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Re: Interesting problems of Permutations and Combinations [#permalink]  01 Feb 2011, 06:07
Bunuel wrote:
mariyea wrote:
Why 7!? There are only six letters to arrange...

Are you talking about Q1, 4th point? Can you please be more specific when asking questions?

4. Permutations of the 7 units {P(S)XXXXS(P)}{X}{X}{X}{X}{X}{X} = 7! = 5040;

There are 7 units: {PXXXXS}, {X}, {X}, {X}, {X}, {X}, {X} --> 6 letters in unit 1, plus 6 units with one letter in each = total of 12 letters.

Yes I am referring to the first q. Sorry about that.

But the q asks for four letters to be placed b/n P and S... and the ways in which the four letters can be arranged is expressed by 4!
How can there be six units that have one letter each?
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Re: Interesting problems of Permutations and Combinations [#permalink]  01 Feb 2011, 06:26
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mariyea wrote:
Yes I am referring to the first q. Sorry about that.

But the q asks for four letters to be placed b/n P and S... and the ways in which the four letters can be arranged is expressed by 4!
How can there be six units that have one letter each?

It seems that you don't understand what the question is asking. It does not ask about the ways 4 lettter can be arranged between P and S.

The question is: in how many ways can the word PERMUTATIONS be arranged SO THAT in each arrangement there are always 4 letters between P and S.

This should be calculated in several steps. Step 4 is dealing with arrangement of 7 units: P and S with 4 letters between them (as required) is one unit {PXXXXS} so we used 6 letters, EACH of the rest 6 letters is a separate unit itself so we have total of 7 units: {PXXXXS}, {X}, {X}, {X}, {X}, {X}, {X} --> 7!.
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Re: Interesting problems of Permutations and Combinations [#permalink]  01 Feb 2011, 06:39
Bunuel wrote:
mariyea wrote:
Yes I am referring to the first q. Sorry about that.

But the q asks for four letters to be placed b/n P and S... and the ways in which the four letters can be arranged is expressed by 4!
How can there be six units that have one letter each?

It seems that you don't understand what the question is asking. It does not ask about the ways 4 lettter can be arranged between P and S.

The question is: in how many ways can the word PERMUTATIONS be arranged SO THAT in each arrangement there are always 4 letters between P and S.

This should be calculated in several steps. Step 4 is dealing with arrangement of 7 units: P and S with 4 letters between them (as required) is one unit {PXXXXS} so we used 6 letters, EACH of the rest 6 letters is a separate unit itself so we have total of 7 units: {PXXXXS}, {X}, {X}, {X}, {X}, {X}, {X} --> 7!.

Bunuel, Bunuel! Thank you so much! I get it now, you're the man!
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Re: Interesting problems of Permutations and Combinations [#permalink]  03 Feb 2011, 02:42
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Bunuel wrote:
chandrun wrote:
Here a couple of problems I encountered. Let me know your views on them. They may not necessarily be of GMAT format. Please share your views, nevertheless.

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Cheers

THEORY:

Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:

$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$.

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.

Back to the original questions:

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

There are 12 letters in the word "PERMUTATIONS", out of which T is repeated twice.

1. Choosing 4 letters out of 10 (12-2(P and S)=10) to place between P and S = 10C4 = 210;
2. Permutation of the letters P ans S (PXXXXS or SXXXXP) = 2! =2;
3. Permutation of the 4 letters between P and S = 4! =24;
4. Permutations of the 7 units {P(S)XXXXS(P)}{X}{X}{X}{X}{X}{X} = 7! = 5040;
5. We should divide multiplication of the above 4 numbers by 2! as there is repeated T.

Hence: $$\frac{10C4*2!*4!*7!}{2!}=25,401,600$$

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

There are 11 letters in the word "MISSISSIPPI ", out of which: M=1, I=4, S=4, P=2.

Total # of permutations is $$\frac{11!}{4!4!2!}$$;
# of permutations with 4 I's together is $$\frac{8!}{4!2!}$$. Consider 4 I's as one unit: {M}{S}{S}{S}{S}{P}{P}{IIII} - total 8 units, out of which {M}=1, {S}=4, {P}=2, {IIII}=1.

So # of permutations with 4 I's not come together is: $$\frac{11!}{4!4!2!}-\frac{8!}{4!2!}$$.

Hope it helps.

What about my approach to PERMUTATIONS?

The word permutations consists of 12 letters.
You can choose P and S in 7*2 ways so there are always 4 numbers between them
P on first, S on fifth ........ P on seventh S on twelwth + reversely (SP)

You are left with 10 letters, 2 of which are the same (TT)

So the complete formula is:

$$7*2*\frac{10!}{2!}=7*10!$$

It makes the same result, but I think is a bit quicker.
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Re: Interesting problems of Permutations and Combinations [#permalink]  03 Feb 2011, 02:48
Expert's post
craky wrote:
Bunuel wrote:
chandrun wrote:
Here a couple of problems I encountered. Let me know your views on them. They may not necessarily be of GMAT format. Please share your views, nevertheless.

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Cheers

THEORY:

Permutations of $$n$$ things of which $$P_1$$ are alike of one kind, $$P_2$$ are alike of second kind, $$P_3$$ are alike of third kind ... $$P_r$$ are alike of $$r_{th}$$ kind such that: $$P_1+P_2+P_3+..+P_r=n$$ is:

$$\frac{n!}{P_1!*P_2!*P_3!*...*P_r!}$$.

For example number of permutation of the letters of the word "gmatclub" is 8! as there are 8 DISTINCT letters in this word.

Number of permutation of the letters of the word "google" is $$\frac{6!}{2!2!}$$, as there are 6 letters out of which "g" and "o" are represented twice.

Number of permutation of 9 balls out of which 4 are red, 3 green and 2 blue, would be $$\frac{9!}{4!3!2!}$$.

Back to the original questions:

1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

There are 12 letters in the word "PERMUTATIONS", out of which T is repeated twice.

1. Choosing 4 letters out of 10 (12-2(P and S)=10) to place between P and S = 10C4 = 210;
2. Permutation of the letters P ans S (PXXXXS or SXXXXP) = 2! =2;
3. Permutation of the 4 letters between P and S = 4! =24;
4. Permutations of the 7 units {P(S)XXXXS(P)}{X}{X}{X}{X}{X}{X} = 7! = 5040;
5. We should divide multiplication of the above 4 numbers by 2! as there is repeated T.

Hence: $$\frac{10C4*2!*4!*7!}{2!}=25,401,600$$

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

There are 11 letters in the word "MISSISSIPPI ", out of which: M=1, I=4, S=4, P=2.

Total # of permutations is $$\frac{11!}{4!4!2!}$$;
# of permutations with 4 I's together is $$\frac{8!}{4!2!}$$. Consider 4 I's as one unit: {M}{S}{S}{S}{S}{P}{P}{IIII} - total 8 units, out of which {M}=1, {S}=4, {P}=2, {IIII}=1.

So # of permutations with 4 I's not come together is: $$\frac{11!}{4!4!2!}-\frac{8!}{4!2!}$$.

Hope it helps.

What about my approach to PERMUTATIONS?

The word permutations consists of 12 letters.
You can choose P and S in 7*2 ways so there are always 4 numbers between them
P on first, S on fifth ........ P on seventh S on twelwth + reversely (SP)

You are left with 10 letters, 2 of which are the same (TT)

So the complete formula is:

$$7*2*\frac{10!}{2!}=7*10!$$

It makes the same result, but I think is a bit quicker.

This approach is also correct.
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Re: Interesting problems of Permutations and Combinations [#permalink]  04 Feb 2011, 05:41
1. In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

PERMUTATIONS:
Total length of the word = 12
Repetitions:
T:2. Rest all letters:1

P,S can occupy following indices in the word
{1,6},{2,7},{3,8},{4,9},{5,10},{6,11},{7,12}

In each of the above positions that P and S occupy, the remaining 10 letters can be arranged in
$$\frac{10!}{2!}$$ ways

Likewise; S,P can occupy following indices:
{1,6},{2,7},{3,8},{4,9},{5,10},{6,11},{7,12}

In each of the above positions that S and P occupy, the remaining 10 letters can be arranged in
$$\frac{10!}{2!}$$ ways

So, total number of arrangements:

$$\frac{7*2*10!}{2!}$$

2. In how many of the distinct permutations of the letters in the word MISSISSIPPI do the 4 I's not come together?

Word MISSISSIPPI
Length: 11
Repetitions:
I:4
S:4
P:2
M:1

Total number of arrangement possible = $$\frac{11!}{4!4!2!}$$ ------------Ist

Now, let's conjoin four I's and treat it as a unique character, say #. We have conjoined all fours in order to symbolize that all I's are adhered together.

So, Now MISSISSIPPI becomes M#SSSSPP
Word: M#SSSSPP
Length: 8
Repetitions:
S:4
P:2
M:1
#:1

The number of ways in which I's come together is:
$$\frac{8!}{2!4!}$$ --------- 2nd

Thus, the number of arrangements in which I's not come together will be the difference between 2nd and Ist

$$\frac{11!}{4!4!2!}-\frac{8!}{2!4!}$$
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Re: Interesting problems of Permutations and Combinations [#permalink]  05 Feb 2011, 03:59
geeeesh bunuel... amazing explanation.
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Permutations Question [#permalink]  05 Jul 2013, 08:35
In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

My approach:

Lets consider P_ _ _ _ S be a single letter.

The total number of letters in PERMUTATIONS = 12

6 letters would be consumed in P_ _ _ _ S, so we have to arrange 7 letters. Hence 7!

Now for the two Ts 7!/2! and for arranging four letters between P and S. (7!/2!) 4!

Is this the right solution? I am not sure if additional multiplication should be done for selecting these four letters from the remaining 10 letters.

Any help is much appreciated.

Thanks,
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Re: Permutations Question [#permalink]  05 Jul 2013, 08:38
Expert's post
ParmarKarishma wrote:
In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?

My approach:

Lets consider P_ _ _ _ S be a single letter.

The total number of letters in PERMUTATIONS = 12

6 letters would be consumed in P_ _ _ _ S, so we have to arrange 7 letters. Hence 7!

Now for the two Ts 7!/2! and for arranging four letters between P and S. (7!/2!) 4!

Is this the right solution? I am not sure if additional multiplication should be done for selecting these four letters from the remaining 10 letters.

Any help is much appreciated.

Thanks,

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Re: In how many ways can the letters of the word PERMUTATIONS be [#permalink]  17 Jul 2014, 21:08
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Re: In how many ways can the letters of the word PERMUTATIONS be [#permalink]  28 Aug 2014, 21:19
In how many ways can the letters of the word PERMUTATIONS be arranged if there are always 4 letters between P and S?
I solved this way:
P_ _ _ _ S _ _ _ _ _ _

10! [ arrange remaining 10 letters]
divided by 2! [ because of 2 T]
multiplied by 2! [ swap P and S]
multiplied by 7 [ P and S can move 6 places ahead]

= 7.10!
Please tell if correct approach?
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Re: In how many ways can the letters of the word PERMUTATIONS be [#permalink]  28 Dec 2015, 00:35
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Re: In how many ways can the letters of the word PERMUTATIONS be   [#permalink] 28 Dec 2015, 00:35
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