Solve the problem.
-Find the number of permutations of the letters in the word DOUBLE.
A)24
B)120
C)5
D)1

Question

Quizplus · Accepted Answer

The number of permutations of n distinct objects is n! (n factorial). In this case, there are 6 distinct letters, so the number of permutations is 6! = 720. However, we have two O's and two U's, which means we are overcounting some of the permutations. Specifically, for each permutation where the two O's are in different positions, we have another permutation where they are swapped (but all other letters remain in the same order). The same is true for the two U's. Therefore, we need to divide the total number of permutations by 2! twice (once for the O's and once for the U's) to correct for the overcounting. This gives us a total of 720 / (2! * 2!) = 720 / 4 = 180 permutations. There are 6 letters in the word DOUBLE, so there are 6 choices for the first letter, 5 choices for the second letter (since one letter has already been chosen), 4 choices for the third letter, 3 choices for the fourth letter, 2 choices for the fifth letter, and only 1 choice for the last letter. Therefore, the total number of permutations is:
6 x 5 x 4 x 3 x 2 x 1 = 720
However, since the two O's and the two E's are indistinguishable, we must divide by the number of ways to arrange those letters, which is 2! for each pair. Thus, the final answer is:
720 ÷ (2! x 2!) = 120.

Solve the Problem. -Find the Number of Permutations of the Letters in the Letters