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?
(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
$={ }^{12} C_{6}-{ }^{8} C_{6}$
$=\frac{12 !}{6 ! 6 !}-\frac{8 !}{6 ! 2 !}$
$=\frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1}-\frac{8 \times 7}{2}$
$=924-28$
= 896
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