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  4. 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

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Q. If the number of circular permutations of 20 letters $P, Q, R, S, T, A, A, A \cdots A(A ' s$ 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

Permutations and Combinations

Solution:

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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 $l, m , n , p , q \in N \cup\{0\}$
$\therefore 2 l+1+2 m +1+2 n +1+2 p +1+2 q +1=15 $
$\therefore l+ m + n + p + q =5 \Rightarrow{ }^9 C _4$
Hence, total number of ways $=4 ! \times{ }^9 C _4=24 \times{ }^9 C _4=12 \cdot{ }^{10} C _5$