If the number of circular permutations of 20 letters P, Q, R, S, T, A, A, A ⋅s A(A ' s are 15 ) such that between two distinct letters there are odd number of alike letters (A's), is k ⋅ 10 C 5 then k is

Question

If the number of circular permutations of 20 letters P, Q, R, S, T, A, A, A ⋅s A(A ' s are 15 ) such that between two distinct letters there are odd number of alike letters (A's), is k ⋅ 10 C 5 then k is (A) 6 (B) 12 (C) 14 (D) 26

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/f42d87a784010214d43fa1c0150e416c-.png" /> Distinct letters are arranged in a circle by 4 ! ways. Let number of alike letters to inserted between distinct letters are 2 l+1,2 m +1,2 n +1,2 p +1,2 q +1 where l, m , n , p , q ∈ N ∪ 0 ∴ 2 l+1+2 m +1+2 n +1+2 p +1+2 q +1=15 ∴ l+ m + n + p + q =5 ⇒ 9 C 4 Hence, total number of ways =4 ! × 9 C 4=24 × 9 C 4=12 ⋅ 10 C 5

Q. If the number of circular permutations of 20 letters P,Q,R,S,T,A,A,A⋯A(A′s are 15 ) such that between two distinct letters there are odd number of alike letters (A's), is k⋅10C5​ then k is

6

12

14

26

Distinct letters are arranged in a circle by 4 ! ways. Let number of alike letters to inserted between distinct letters are 2l+1,2m+1,2n+1,2p+1,2q+1 where l,m,n,p,q∈N∪{0} ∴2l+1+2m+1+2n+1+2p+1+2q+1=15 ∴l+m+n+p+q=5⇒9C4​ Hence, total number of ways =4!×9C4​=24×9C4​=12⋅10C5​

Q. If the number of circular permutations of 20 letters $P, Q, R, S, T, A, A, A \dots A (A^{'} s$ are 15 ) such that between two distinct letters there are odd number of alike letters (A's), is $k \cdot^{10} C_{5}$ then $k$ is