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Olive is creating a five-digit code using the digits 0 through 9. How

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Joined: 01 May 2016
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Olive is creating a five-digit code using the digits 0 through 9. How  [#permalink]

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14 Mar 2018, 05:21
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Olive is creating a five-digit code using the digits 0 through 9. How many different codes can she create with exactly two prime digits if no digits can be repeated?

A. 252

B. 3,120

C. 3,456

D. 14,400

E. 30,240

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Posts: 57155
Re: Olive is creating a five-digit code using the digits 0 through 9. How  [#permalink]

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15 Mar 2018, 00:35
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Tridhipal wrote:
Olive is creating a five-digit code using the digits 0 through 9. How many different codes can she create with exactly two prime digits if no digits can be repeated?

A. 252

B. 3,120

C. 3,456

D. 14,400

E. 30,240

The code will be any combination of the following: {P1} {P2} {X} {Y} {Z}, where P1 and P2 are prime digits, and X, Y and Z are other digits (all distinct).

There are four single-digit primes: 2, 3, 5, and 7 and six other digits: 0, 1, 4, 6, 8, and 9.

The number of pairs of primes, P1 and P2, out of four primes is 4C2 = 6;
The number of triplets of non-primes, X, Y and Z, our of six is 6C3 = 20.

Therefore, 6*20*5! = 14,400 (multiplying by 5! to account for different arrangements of 5 digits {P1} {P2} {X} {Y} {Z}).

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Re: Olive is creating a five-digit code using the digits 0 through 9. How  [#permalink]

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19 Mar 2018, 06:40
Tridhipal wrote:
Olive is creating a five-digit code using the digits 0 through 9. How many different codes can she create with exactly two prime digits if no digits can be repeated?

A. 252

B. 3,120

C. 3,456

D. 14,400

E. 30,240

From 0 through 9, the prime digits are 2, 3, 5, and 7, and thus the non-prime digits are 0, 1, 4, 6, 8, and 9.

The number of ways to select the 2 prime digits is 4C2 = (4 x 3)/2 = 6.

The number of ways to select the 3 non-prime digits is 6C3 = (6 x 5 x 4)/(3 x 2) = 20.

Therefore, there are 6 x 20 = 120 ways to create these codes if order doesn’t matter. However, for each code created, the order matters. For example, the code 27014 is different from 70412, 42017, etc. For each of 120 codes, since there are 5 different digits, there are 5! = 120 ways to arrange them. Thus the total number of codes can be created is:

120 x 120 = 14,400

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Joined: 28 May 2012
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Re: Olive is creating a five-digit code using the digits 0 through 9. How  [#permalink]

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26 Feb 2019, 01:52
Why do you use the combinations formula here?

I understand that there are 4 primes and 6 non-primes. Therefore, I thought the path would be as follows: 4 (options for primes) * 3 (options for primes - 1) * 6 (options for non-primes) * 5 (options for non-primes -1 ) * 4 (options for non-primes -2). What is wrong with this logic?
Intern
Joined: 20 Feb 2019
Posts: 1
Re: Olive is creating a five-digit code using the digits 0 through 9. How  [#permalink]

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27 Feb 2019, 08:55
Synchrony wrote:
Why do you use the combinations formula here?

I understand that there are 4 primes and 6 non-primes. Therefore, I thought the path would be as follows: 4 (options for primes) * 3 (options for primes - 1) * 6 (options for non-primes) * 5 (options for non-primes -1 ) * 4 (options for non-primes -2). What is wrong with this logic?

You didn't take into account the way of arranging the digits!
Re: Olive is creating a five-digit code using the digits 0 through 9. How   [#permalink] 27 Feb 2019, 08:55
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