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How many different license passwords?
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Updated on: 21 Sep 2023, 04:22
Updated on: 21 Sep 2023, 04:22
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Question Stats:
50% (02:16) correct
50% (01:58) wrong based on 2 sessions
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How many different license passwords can one make if said password must contain exactly 6 characters, two of which are distinct numbers, another of which must be an upper-case letter, and the remaining 3 can be any digit or letter (upper- or lower-case) such that there are no repetitions of any characters in the password?
A. 456426360
B. 365580800
C. 619652800
D. 219
E. 231
A. 456426360
B. 365580800
C. 619652800
D. 219
E. 231
Re: How many different license passwords?
[#permalink]
21 Sep 2023, 04:20
21 Sep 2023, 04:20
So, The way I am solving is -
ways to select two distinct numbers from 10 = 10C2
ways to select an uppercase letter = 26C1
Ways to select three characters from 59 = 59C3
Now, number of choices = 10C2 * 26C1 * 56C3
Since order matters, we need to multiply by 6!
Thus, final answer = 10C2 * 26C1 * 59C3 * 6! = 27,385,581,600
But, the correct answer is 456426360. Where am I going wrong?
ways to select two distinct numbers from 10 = 10C2
ways to select an uppercase letter = 26C1
Ways to select three characters from 59 = 59C3
Now, number of choices = 10C2 * 26C1 * 56C3
Since order matters, we need to multiply by 6!
Thus, final answer = 10C2 * 26C1 * 59C3 * 6! = 27,385,581,600
But, the correct answer is 456426360. Where am I going wrong?
Re: How many different license passwords?
[#permalink]
21 Sep 2023, 04:22
21 Sep 2023, 04:22
Expert Reply
OE
Begin by considering the three "hard and fast conditions" - the digits and the one upper-case letter. For the first number, you will have 10 choices and for the second 9 (since you cannot repeat). For the captial letter, you have 26 choices. Thus far, your password has 10 * 9 * 26 possible combinations.
Now, given your remaining options, you have 8 digits, 25 upper-case letters, and 26 lower-case letters (i.e. 59 possible choices). Since you cannot repeat, you will thus have for your remaining choices 59, 58, and 57 possibilities.
Putting all of this together, you have: 10 * 9* 26 *59 * 58 *57 or 456426360 choices.
Begin by considering the three "hard and fast conditions" - the digits and the one upper-case letter. For the first number, you will have 10 choices and for the second 9 (since you cannot repeat). For the captial letter, you have 26 choices. Thus far, your password has 10 * 9 * 26 possible combinations.
Now, given your remaining options, you have 8 digits, 25 upper-case letters, and 26 lower-case letters (i.e. 59 possible choices). Since you cannot repeat, you will thus have for your remaining choices 59, 58, and 57 possibilities.
Putting all of this together, you have: 10 * 9* 26 *59 * 58 *57 or 456426360 choices.
