BTGModeratorVI wrote: ↑Mon Apr 13, 2020 3:41 pm
The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter?
A) 12
B) 18
C) 24
D) 36
E) 48
Answer:
D
Source: Official guide
This is a permutation problem because the order of the letters matters. Let’s first determine in how many ways we can arrange the letters. Since there are 2 repeating I’s, we use the indistinguishable permutations formula, and we see that we can arrange the letters in 5! / 2! = 120 / 2 = 60 ways.
We also have the following equation:
60 = (number of ways to arrange the letters with the I’s together) + (number of ways without the I’s together).
Let’s determine the number of ways to arrange the letters with the I’s together. Note that we treat the two I’s as a single item, so we are in essence finding the number of permutations of 4 items.
We have: [I-I] [D] [G] [T]
We see that with the I’s together, we have 4! = 24 ways to arrange the letters.
Thus, the number of ways to arrange the letters without the I’s together (i.e., with the I’s separated) is 60 - 24 = 36.
Answer: D
