How many 4-permutations are there of the set <mo fence="false" stretchy="false">{ A , B ,

cazinskup3

cazinskup3

Answered question

2022-06-12

How many 4-permutations are there of the set { A , B , C , D , E , F } if whenever A appears in the permutation, it is followed by E?
Case 1: when A does not appear in the 4-permutation
5 4 3 2 = 120
Case 2: when A does appear in the 4-permutation. Then E should follow A. Let X stand for AE, now we should consider 3-permutations, since X occupies two spaces. Since X stands for two elements, we consider the number of elements to be 5.
5 4 3 = 60
Thus, number of permutations of the set { A , B , C , D , E , F } where if whenever A appears in the permutation, it is followed by E is 120 + 60 = 180.

Answer & Explanation

Ryan Newman

Ryan Newman

Beginner2022-06-13Added 26 answers

Step 1
You handled case 1 correctly. Your second approach to case 2 is correct. In your first approach to case 2, there are not five elements left to choose once you include AE, just two of the remaining four elements.
For case 2, if E immediately follows A, there are three places to place the block AE since it must begin in one of the first three positions, four ways to place one of the remaining four letters in the first open position, and three ways to place one of the remaining three letters in the last open position. Hence, there are 3 4 3 = 36 admissible arrangements in case 2.
Step 2
Alternatively, there are ( 4 2 ) ways to select the other two letters which appear with the block AE and 3! ways to arrange the block AE and the other two selected letters, so there are ( 4 2 ) 3 ! = 36 admissible permutations in case 2.
Hence, there are a total of 120 + 36 = 156 admissible 4-permutations of the letters { A , B , C , D , E , F } if whenever A appears in the permutation, it is followed by E.

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