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How many 4-letter words can be made from the letters of the word PULLEYS? Explain your answer.

There are two mutually exclusive cases by which four letter-word can be arranged from the word PULLEYS. Let n(A) be the number of arrangements of 4-letter word with all different letters. Let n(B) be the number of arrangements that contain two L's and two other letters.

Case A:

There are C(6, 4) ways to choose subsets of four different letters. Each of these subsets has the length of four and can generate 4! sequences of letters. Therefore, n(A) = C(6, 4) x 4! = 360.

Case B:

There are C(2, 2) X C(5, 2) subsets that contain two L's and two other letters. The letters in each subset can be arranged in: C(4, 2) X C(2, 1) X C(1,1) different ways. Therefore, n(B) = C(2, 2) X C(5, 2) X C(4, 2) X C(2, 1) X C(1,1) = C(5, 2) X 4!/2! = 120

Therefore, the total number of four-letter words: n(A) + n(B) = 480

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# Homework Help: 4-letter words can be made from pulleys?

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