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# The security gate at a storage facility requires a fivE

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The security gate at a storage facility requires a fivE [#permalink]

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23 Feb 2011, 22:05
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The security gate at a storage facility requires a five-digit lock code. If the lock code must consist only of digits from 1 through 7, inclusive, with no repeated digits, and the first and last digits of the code must be odd, how many lock codes are possible?

A. 120
B. 240
C. 360
D. 720
E. 1440
[Reveal] Spoiler: OA

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24 Feb 2011, 04:16
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IndigoIntentions wrote:
The security gate at a storage facility requires a five-digit lock code. If the lock code
must consist only of digits from 1 through 7, inclusive, with no repeated digits, and
the first and last digits of the code must be odd, how many lock codes are possible?

a)120
b)240
c)360
d)720
e)1440

X-X-X-X-X --> there are 4 odd digits from 1 to 7, inclusive thus options for the first and the last X's are: 4-X-X-X-3. Other X's can take following values: 4-5-4-3-3 --> 4*5*4*3*3=720.

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24 Feb 2011, 04:21
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Digit #1: has to be odd => 4 possibilities (1,3,5,7)

Digit #5: has to be odd => 3 possiblities left (one possibility has already been used for digit #1)

Digit #2: any of the 7 possible digits (1,2,...7) less the 2 that have already been used for Digits #1 & #5 => 5 possibilities

Digit #3: any of the 7 possible digits (1,2,...7) less the 3 that have already been used for Digits #1, #2 & #5 => 4 possibilities

Digit #4: any of the 7 possible digits (1,2,...7) less the 4 that have already been used for Digits #1, #2, #3 & #5 => 3 possiblities

Thus, the number of possiblities is 4*3*5*4*3 = 720

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24 Feb 2011, 04:57
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4 * 3 * 5!/2!

= 12 * (5 * 4 * 3* 2!)/2!

= 12 * 60 = 720

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18 Apr 2013, 22:17
Bunuel wrote:
IndigoIntentions wrote:
The security gate at a storage facility requires a five-digit lock code. If the lock code
must consist only of digits from 1 through 7, inclusive, with no repeated digits, and
the first and last digits of the code must be odd, how many lock codes are possible?

a)120
b)240
c)360
d)720
e)1440

X-X-X-X-X --> there are 4 odd digits from 1 to 7, inclusive thus options for the first and the last X's are: 4-X-X-X-3. Other X's can take following values: 4-5-4-3-3 --> 4*5*4*3*3=720.

Hi bunuel,

I have a small doubt , why not we are considering that the first and last digit can occur in 2 ways and the middle 3 digits can occur in 6 ways.

for example middle 3 digits can be arranged in 3! ways among themselves after selecting 5c1 4c1 3c1 .

My question may be stupid, please correct my doubt.

Thanks

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19 Apr 2013, 03:28
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tarunjagtap wrote:
Bunuel wrote:
IndigoIntentions wrote:
The security gate at a storage facility requires a five-digit lock code. If the lock code
must consist only of digits from 1 through 7, inclusive, with no repeated digits, and
the first and last digits of the code must be odd, how many lock codes are possible?

a)120
b)240
c)360
d)720
e)1440

X-X-X-X-X --> there are 4 odd digits from 1 to 7, inclusive thus options for the first and the last X's are: 4-X-X-X-3. Other X's can take following values: 4-5-4-3-3 --> 4*5*4*3*3=720.

Hi bunuel,

I have a small doubt , why not we are considering that the first and last digit can occur in 2 ways and the middle 3 digits can occur in 6 ways.

for example middle 3 digits can be arranged in 3! ways among themselves after selecting 5c1 4c1 3c1 .

My question may be stupid, please correct my doubt.

Thanks

Consider this: two digit code XX, each digit must be distinct and can be 1, 2 or 3.

First digit can take 3 values (1, 2, or 3) and the second can take 2 values, total 3*2=6 codes:
12
13
21
23
31
32.

Similarly, for the original question: the first digit can take 4 values and the last digit can take 3 values, total 4*3. The same way for the middle 3 digits: the second digit can take 5 values (7 minus two we already used for first and last), the third 4 and the fourth 3.

Total: (4*3)*5*4*3=720.

Hope it's clear.
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Re: The security gate at a storage facility requires a fivE [#permalink]

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24 Apr 2013, 23:13
IndigoIntentions wrote:
The security gate at a storage facility requires a five-digit lock code. If the lock code must consist only of digits from 1 through 7, inclusive, with no repeated digits, and the first and last digits of the code must be odd, how many lock codes are possible?

A. 120
B. 240
C. 360
D. 720
E. 1440

1. There is one large group i.e., digits 1 through 7, from which 5 digits should be selected.
2. The first and the last digits have a constraint that they should be odd. There is also a general constraint that each digit should be different.
3. Considering the above constraints the first and the last digits can be selected in $$4P2$$ ways. 4 because there are 4 odd digits out of which they can be selected and because there should be no repetition it is $$4P2$$ ways for those 2 positions.
4. The second digit cannot have the same digit as the first and the last digit. Therefore it can be chosen out of 7-2=5 digits. Similarly the third digit cannot have the same digit as the first, second and last. So it can be chosen in 4 ways and the fourth digit similarly in 3 ways.
5. So the number of lock codes possible is $$4P2 * 5* 4*3 = 720$$

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Re: The security gate at a storage facility requires a fivE [#permalink]

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20 Jun 2013, 17:23
We have four odd numbers in this set 1, 3, 5, 7

First digit is 4C1 = 4
Last digit is 3C1 = 3 (we have to exclude the odd digit selected in the previous step.

mid digits are 3 of the remaining 5 digits (2, 4 , 6, in addition to two not selected odd digits) for the 2nd, 3rd and 4th digits in the code

mid digits 5C3 = 60

4*60*3= 720

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Re: The security gate at a storage facility requires a fivE [#permalink]

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25 Aug 2013, 03:56
Odd digits between 1 and 7
1 3 5 7 ==> 4
first and last place being fixed, can be filled with Odd number is
4* 5 *4* 3* 3

Total 720 combinations

Ans: D

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Re: The security gate at a storage facility requires a fivE [#permalink]

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Re: The security gate at a storage facility requires a fivE [#permalink]

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02 Jul 2016, 16:52
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Re: The security gate at a storage facility requires a fivE [#permalink]

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05 Oct 2016, 07:04
Hi Bunuel,
Small clarification is required on this question.

Lets say.. First and last digits should be odd numbers . so it can be done in 5 ways (First) & 4 Ways (Last). Now, Since 5 digits are remaining for 3 places, ( 5C3 X 3!) can be done.... Finally answer is 360 ( can you explain?)

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Re: The security gate at a storage facility requires a fivE [#permalink]

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09 Oct 2016, 13:00
rajendra00 wrote:
Hi Bunuel,
Small clarification is required on this question.

Lets say.. First and last digits should be odd numbers . so it can be done in 5 ways (First) & 4 Ways (Last). Now, Since 5 digits are remaining for 3 places, ( 5C3 X 3!) can be done.... Finally answer is 360 ( can you explain?)

Dear rajendra00

Quote:
The security gate at a storage facility requires a five-digit lock code.

Attachment:

Places.PNG [ 1.21 KiB | Viewed 1809 times ]

Quote:
If the lock code must consist only of digits from 1 through 7

So we have the following digits -
Attachment:

Digits.PNG [ 1.67 KiB | Viewed 1809 times ]

Quote:
first and last digits of the code must be odd,

So, We have 4 digits to be filled in 2 places that can be done as -
Attachment:

Odd Digit.PNG [ 1.7 KiB | Viewed 1808 times ]

Now we are left with 5 digits , ( 2 Odd Digits and 3 Even Digits ) and we now need to fill up 3 places using 5 digits that can be done as -
Attachment:

remaining Digits.PNG [ 1.24 KiB | Viewed 1810 times ]

Now, Final Picture is
Attachment:

Final.PNG [ 1.9 KiB | Viewed 1809 times ]

So, Total Number of ways is 4*5*4*3*3 = 720 ways..

Hence answer will be (D) 720..

Hope this helps.
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The security gate at a storage facility requires a fivE [#permalink]

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03 Jul 2017, 06:05
IndigoIntentions wrote:
The security gate at a storage facility requires a five-digit lock code. If the lock code must consist only of digits from 1 through 7, inclusive, with no repeated digits, and the first and last digits of the code must be odd, how many lock codes are possible?

A. 120
B. 240
C. 360
D. 720
E. 1440

Security gate requires a five digit lock code.
Lock code can take digits from 1 to 7 inclusive i.e. 1,2,3,4,5,6,7... Odd nos. among these nos are 1,3,5,7,total 4 in numbers.

Lets make boxes below to check in how many digits can be placed at the 5 different places of the lock code.

| 4ways (only odd allowed) | 5ways | 4ways | 3ways | 3ways (only odd allowed) |

Let me explain this..
At the first place we can place any of the 4 digits.
At the last place we can place any of 3 digits. (as digits can't be repeated.)

Now there are 5 digits left.
So, at the 2nd place we can place any of 5 digits.
at the 3rd place we can place any of 4 digits.
at the 4th place we can place any of 3 digits.

So, total number of lock codes possible = 4 * 5 * 4 * 3 * 3 = 80 * 9 = 720.

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Re: The security gate at a storage facility requires a fivE [#permalink]

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04 Jul 2017, 04:54
Hi,

I always get confused as to when to use the arrangement formula, i.e. n!, and when to use the combination formula, i.e. nCp.

In the given question I understood the selection of the 1st and last place.

My doubt is for the middle 3 places. Now since we have to 'choose 3 digits out of 5' to fill the middle 3 places, why can't we do 5C3?

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Re: The security gate at a storage facility requires a fivE [#permalink]

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05 Jul 2017, 06:27
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ameyaprabhu wrote:
Hi,

I always get confused as to when to use the arrangement formula, i.e. n!, and when to use the combination formula, i.e. nCp.

In the given question I understood the selection of the 1st and last place.

My doubt is for the middle 3 places. Now since we have to 'choose 3 digits out of 5' to fill the middle 3 places, why can't we do 5C3?

Use arrangement formula n! only when u need to arrange the things like.

I am taking a very simple example for this .
If u want to make an arrangement of numbers 3,4,5 = 3! = 6
So lets see it by making the arrangement (345),(354),(534),(543)(453),(435) = Total 6 .

nCp is used when u have to make selection . For e.g. u need to select the 2 numbers from the three given numbers 3,4,5, = 3C2 = 3 ways.

Coming to the given question.
U have already understood the the first and last numbers,

Now for the middle places, We have to choose 3 digit out of 5 to fill the places. So it can be done in 5 x4 x3 ways = 60 ways

It can also be done in the way you suggested. Lets do it that way. So according to you choose 3 digits out of 5 to fill the middle 3 places = 5C3 = 5!/3!/2!
But these 3 numbers which you have selected out of 5 can also be arranged among themselves in 3! ways as explained in the example of 3,4,5. So you need to multiply your section with 3!. So the total ways become 5!/3!/2! * 3! = 5!/2! = 5*4*3 = 60 ways.. So answer comes to same ... either your way or my way.. Its just you need to think completely. Don't forget to appreciate with kudos if you like my answer.

I hope it solves your issue. Practice more questions to understand. With practice you will find that you don't need any formula and you can just calculate either way...
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Re: The security gate at a storage facility requires a fivE [#permalink]

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05 Jul 2017, 08:13
Let x-x-x-x-x be the 5-digit code.
No. of possibilities for each digit = 4-5-4-3-3 = 4*5*4*3*3 = 720 possibilities. Ans - D.
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Re: The security gate at a storage facility requires a fivE [#permalink]

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05 Jul 2017, 21:23
Thanks for the detailed explanation. Knowing the reasoning helps to apply the formula better.

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Re: The security gate at a storage facility requires a fivE   [#permalink] 05 Jul 2017, 21:23
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