Answer to Question #191575 in Discrete Mathematics for Leia

Question

(a) How many code words over a, b, c, d of length 20 contain exactly 10 a’s?

(b) How many contain exactly 10 a’s and 5b’s.

Answer 1

(a) Remaining 20-10=10 places to be filled with b,c,d. Each of these places can be filled in 3 ways. Hence total number of ways of filling is $3^{10}.$ Out of 20 places 10 places can be chosen in $(\begin{matrix} 20\\ 10 \end{matrix})$ ways. Hence answer is $3^{10}$$(\begin{matrix} 20\\ 10 \end{matrix})$ .

(b) Here as before we need to fill 5 places with c and d. This can be done in $2^5$ ways. The 5 places can be chosen in $(\begin{matrix} 20\\ 5 \end{matrix})$ ways. Now a and b with first one 10 times repeated and b 5 times repeated can be done in $\frac{15!}{10!5!}$ ways. Hence no of words is $2^5 (\begin{matrix} 20\\ 5 \end{matrix})$ $\frac{15!}{10!5!}$ ways.