Hi All,
We're told that a code is to be made by arranging 7 letters: three of the letters used will be the letter A, two of the letters used will be the letter B, one of the letters used will be the letter C, and one of the letters used will be the letter D - and that there is only one way to present each letter. We're asked for the number of different codes that are possible. This question is based on a specific Permutation rule. I'm going to start with a much simpler example of the rule before applying the rule to this question.
Imagine if you had just 3 letters: two As and one B and the A's can only be written in one way (re: they're identical). How many different 3-letter codes could you form?
IF... we actually had 3 DIFFERENT letters - for example: A, B and C, then there would be (3)(2)(1) = 3! = 6 possible codes. You could also write them all out:
ABC
ACB
BAC
BCA
CAB
CBA
With two identical letters though, the number of possible codes decreases. Instead of 6, it's....
AAB
ABA
BAA
...THREE possible codes. From a math-standpoint, the way that we reduce from the number 6 is to DIVIDE by the factorial of whatever letter is duplicated. Here that would be...
3!/2! since there are two As. that gives us (3)(2)(1) / (2)(1) = 3 possible codes
Using that same rule with this prompt, we have a 7-letter code, but three identical As and two identical Bs. Thus, we divide 7! by 3! and 2! This gives us...
(7)(6)(5)(4)(3)(2)(1) / (3)(2)(1)(2)(1) = (7)(6)(5)(4) / (2)(1) = (7)(6)(5)(2) = 420 possible codes
Final Answer:
GMAT assassins aren't born, they're made,
Rich