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# A security company can use number 1 -7 to create a 5 digit

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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
720 is the answer.

They use a "slot technique":

Ie. Spaces _,_,_,_,_

Space one has 4 options (1,3,5,7) and the last space has 3 (the 3 non-chosen odds from space one)

So we're sitting at 4,_,_,_,3

So between the first and last number been chosen 2 out of the 7 possibilites have been used leaving us with

4*5*4*3*3=720 combos

Ahhhhh, my problem on the related question is that I assumed the numbers were 1-9 when it was actually 0-9.

FRIG!!!

Thanks for the help guys!!
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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
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It's a relatively easy permutation problem that includes two restrictions. The order here matters because logically the code will 76543 will be different from 76534, so we possibilities for choice are high too.
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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
TheSituation wrote:
A security company can use number 1 -7 to create a 5 digit lock combo. No numbers can be repeated and the first and last digit must be odd. How many unique lock combos can be created?

Ok, in the MGMAT guide on combinatronics they have two similar problems with & IMO they use two routes in solving them. This is frustrating me. I'm counting on the resident experts here at GMATclub to supply me with the defacto solution.

This is a good question. Can someone please add options and provide OA?
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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
Numbers to be used: 1, 2, 3, 4, 5, 6, 7

Lock: 5 digits: _ _ _ _ _

Condition: No numbers can be repeated and the first and last digit must be odd.

Odd numbers: 1, 3, 5, 7.

$$1^{st}$$ place has 4 options and last place [$$(5)^{th}$$] will have 3 options.

=> $$2^{nd}$$ place will have: 5 options

=> $$3^{rd}$$ place will have: 4 options

=> $$4^{th}$$ place will have: 3 options

Total ways: 4 * 5 * 4 * 3 * 3 = 720

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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
I Always like to break these "Decisions" in Stages to make it clearer and easier to understand:

Available Digits: 1 --- 2 --- 3 -- 4--- 5 -----6 --- 7

Stage 1) We need to pick 2 of the 4 Odd Digits to be placed in the 1st and Last Spots - and then Arrange those 2 Selected Digits

-1- No. of Ways we can Choose 2 Odd Digits out of 4
4 - Choose - 2 = 4! / (2!2!) = 6 Ways

AND

-2- For Each Possible Group of 2 Digits, we can Arrange these 2 Digits between the 1st and 5th Spot in --- 2! Ways = 2 Ways

-------(6) * (2) = 12 Ways to Arrange the 1st and 5th Odd Digits

(Stage 2) For Each of the 12 Arrangements above, we will have 5 Digits remaining from which to Choose. We need to Choose 3 ---- and then find How Many Ways we can Arrange those 3 Chosen

-1- "5 - choose - 3" = 5! / (3!2!) = 10 Ways

AND

-2- For Each Group of 3 Digits Chosen, in how many ways can we Arrange the 3 Digits?

3! Ways = 6 Ways

-----(10) * (6) = 60 Ways to Choose and Arrange the Other 3 Digits

Fundamental Counting Principle: Stage 1 and Stage 2 are Events that happen one after another to form 1 Whole Scenario of a 5 Digit Code

(12 Ways to Choose and Arrange the 1st and 5th Digits) * (60 Ways to Choose and Arrange the Other 3 Digits) =

12 * 60 = 720 Different Codes

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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
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A security company can use number 1 -7 to create a 5 digit lock combo. No numbers can be repeated and the first and last digit must be odd. How many unique lock combos can be created?

Odd numbers in the list :1,3,5,7

Conditions given:No numbers can be repeated and the first and last digit must be odd.

First digit can be filled by 4 ways as there are 4 odd numbers in the list.

Last digit can by filled by 3 ways because its given that no numbers can be repeated.

Out of 7 digits , two of them are used and we are left with five digits.

Second place can be filled by 5 ways .
Third and fourth places can by 4 and 3 ways respectively.

Total no of arrangements = 4 * 3 * 5 * 4 * 3 = 720 ways.
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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
Bunuel this one popped up in the quiz and once again, there was no answer choice...
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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
Khwarizmi wrote:
Bunuel this one popped up in the quiz and once again, there was no answer choice...

There are 0.01% questions with no OA's. We are working on it. Thank you!
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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
1,2,3,4,5,6,7
1st and last digit={1,3,5,7} =4C2 2!
Rest all digits={o,o, 2,4,6} =5C3 3!

Total combos=12*10*6
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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
In this case, I made it like this:

$$4C2 * 5 * 4 * 3 = 360$$

Considering that the first and last digits must be fixed odd numbers (4 out of 7).

What's my error?

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A security company can use number 1 -7 to create a 5 digit [#permalink]
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Lodz697 wrote:
In this case, I made it like this:

$$4C2 * 5 * 4 * 3 = 360$$

Considering that the first and last digits must be fixed odd numbers (4 out of 7).

What's my error?

Bunuel

Posted from my mobile device

The two odd digits at the front and back can themselves be arranged in 2 ways
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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
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TheSituation wrote:
A security company can use number 1 -7 to create a 5 digit lock combo. No numbers can be repeated and the first and last digit must be odd. How many unique lock combos can be created?

A. 120
B. 240
C. 360
D. 480
E. 720

Ok, in the MGMAT guide on combinatronics they have two similar problems with & IMO they use two routes in solving them. This is frustrating me. I'm counting on the resident experts here at GMATclub to supply me with the defacto solution.

We can use the fundamental counting principle here. We have 5 empty slots in the password. We have 7 digits, 4 odd and 3 even.

O __ __ __ O

To fill in the first blank with an odd digit, we have 4 options.
To fill in the last blank with an odd digit, we have 3 options. (1 odd digit cannot be used)
To fill in the second blank, we have 5 options (2 leftover odd digits and 3 even digits)
To fill in the third blank, we have 4 options.
To fill in the fourth blank, we have 3 options.

Total ways = 4 * 3 * 5 * 4 * 3 = 720

This counting principle is discussed here: https://youtu.be/LFnLKx06EMU
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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
TheSituation wrote:
A security company can use number 1 -7 to create a 5 digit lock combo. No numbers can be repeated and the first and last digit must be odd. How many unique lock combos can be created?

A. 120
B. 240
C. 360
D. 480
E. 720

Ok, in the MGMAT guide on combinatronics they have two similar problems with & IMO they use two routes in solving them. This is frustrating me. I'm counting on the resident experts here at GMATclub to supply me with the defacto solution.

I think my approach will be hectic if the number set is large but here is how I did it.
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Re: A security company can use number 1 -7 to create a 5 digit [#permalink]
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