Question

Question # 4.
(a) Determine the number of ways to rearrange the letters in the word QUESTION.

(b) Determine the number of ways to rearrange the letters in the word BOOKKEEPERS.

(c) Determine the number of ways to rearrange the letters in the word SUCCESSFULLY, assuming that all the Ss are kept together, and the E and F are not side-by-side

Answer 1

(a) QUESTION has 8 distinct letters. We know that n distinct objects can be arranged on a straight line in n! ways. Hence, they can be arranged in 8! = 40320 ways

(b) When letters start to repeat, we need to divide, for each of the repeated letters, by the factorial of the number of times that they repeat.

Here the letter is BOOKKEEPERS - total 11 letters, but out of them there are 2 O's, 2 K's and 3 E's. Hence, the number of ways to rearrange the letters in the word BOOKKEEPERS is

(c) SUCCESSFULLY - Total 12 letters out of which there are 3 S's, 2 U's, 2 C's and 2 L's

Make a "bundle" of all the 3 S's, so that they are always together. Now we have 10 objects, the 9 letters and a bundle of all S's together

Since E and F are not side-by-side so for the time being keep them separate and just arrange the other 8 objects

These 8 objects also have thre pair of letters, each repeated twice. Hence number of ways to arrange them is

Now for each of these 5040 ways that we have arranged these 8 objects, the E and F need to be placed as well. But they only neeed to be placed in the 9 gaps that get created, when these 8 objects are arranged. By placing them in these gaps, we are ensuring that they are not together

So we need to choose 2 out of 9 gaps, which can be done in C(9, 2) = 36 ways.

Since these two methods happen in succession, so we need to multiple to get toal arrangements. Hence,