Question

From 4 officers and 8 jawans in how many ways can 6 be chosen (i) to include exactly one officer (ii) to include at least one officer?

Answer 1

(i) From 4 officers and 8 jawans, 6 need to be chosen. Out of them, 1 is an officer.
Required number of ways = ${}^4{C}_1\times {}^8{C}_5=4\times \frac{8!}{5!3!}=4\times \frac{8\times 7\times 6\times 5!}{5!\times 6}=224$

(ii) From 4 officers and 8 jawans, 6 need to be chosen and at least one of them is an officer.
Required number of ways = Total number of ways $-$ Number of ways in which no officer is selected
$$\begin{gathered} {=}^{12}{C}_6{-}^8{C}_6 \\ =\frac{12!}{6!6!}-\frac{8!}{6!2!} \\ =\frac{12\times 11\times 10\times 9\times 8\times 7}{6\times 5\times 4\times 3\times 2\times 1}-\frac{8\times 7}{2} \\ =924-28 \\ =896 \end{gathered}$$