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Find the number of ways in which a commi...

Find the number of ways in which a committee of 6 members can be formed out of 4 officers and 8 jawans, if there are at least 2 officer in the committee.

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The committee can be formed in the following ways:
(1) 2 offers + 4 jawans.
No. of ways of selection `= .^(4)C_(2) xx .^(8)C_(4)`
(ii) 3 offers +3 jawans.
No. of ways of selection `= .^(4)C_(3) xx .^(8)C_(3)`
(iii) 4 officers +2 jawans.
No. of ways of selection `= .^(4)C_(4) xx .^(8)C_(2)`
The committee can be formed in any one way, therefore from addition principle
Required ways `= .^(4)C_(2) xx^(8)C_(4) +^(4)C_(3) xx^(8)C_(3)+^(4)C_(4) xx^(8)C_(2)`
`= 6 xx 70 +4 xx 56 +1 xx 28`
`= 420 +224 +28 = 672`
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