Question: 7. A string that is obtained by rearranging the letters of the word BOOGER is called awesome, if the string does not contain the substring OO. Thus, GEOROB is awesome, whereas GREOOB is not awesome. What is the number of awesome strings?
Answer: (6 * 5C2 * 3 * 2 * 1)
Attempt: What I did was calculate the number of ways the string contains the substring OO, which I found to be 5!, I treated OO as one letter. Then I subtracted it from the total number of possible substrings 6!.I'm not sure where I went wrong, would appreciate any help.
The answer is $240$, which (as the comments indicate) is actually the result the OP's book provides. Here are two ways to obtain that answer:
First, to apply the method the OP uses, we need only fix one minor error—there are not $6! = 720$ different permutations of BOOGER, but only $6!/2 = 360$. The reason is that the two O's are not distinguishable. If we permute BO$_1\!$O$_2\!$GER, there are $720$ different sequences, but in ordinary text, there is no difference between BO$_1\!$O$_2\!$GER and BO$_2\!$O$_1\!$GER, and in general, each permutation of BO$_1\!$O$_2\!$GER has an indistinguishable "twin". Therefore, the correct answer is $6!/2 - 5! = 360-120 = 240$.
A second approach is to first decide where to put the two O's. There are ten possibilities: OXOXXX, OXXOXX, OXXXOX, OXXXXO, XOXOXX, XOXXOX, XOXXXO, XXOXOX, XXOXXO, and XXXOXO. The remaining four spots (marked by X's above) can be replaced by the four remaining letters B, G, E, and R in $4! = 24$ different ways. So the number of awesome permutations is $10 \times 24 = 240$.