From 4 officers and 8 jawans in how many ways can 6 be chose To include exactly one officer To include at least one officer?

Explanation

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 `
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