Question

From 4 officers and 8 jawans in how many ways can 6 be chosen. to include at least one officer?

Answer 1

 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

\[=^{12} C_6 -^8 C_6 \]
\[ = \frac{12!}{6! 6!} - \frac{8!}{6! 2!} \]
\[ = \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7}{6 \times 5 \times 4 \times 3 \times 2 \times 1} - \frac{8 \times 7}{2} \]
\[ = 924 - 28\]
\[ = 896\]