From 4 officers and 8 jawans in how many ways can 6 be chosen

(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

$={ }^{12} C_{6}-{ }^{8} C_{6}$

$=\frac{12 !}{6 ! 6 !}-\frac{8 !}{6 ! 2 !}$

$=\frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1}-\frac{8 \times 7}{2}$

$=924-28$ 

= 896