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A committee of 5 is to be formed out of 6 gents and 4 ladies. In how many ways this can be done when
 atleast two ladies are included.
 atmost two ladies are included.
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Solution
(i) A committee of 5 is to be formed.
Possibilities  Ladies (4)  Gents (6)  Combinations 
1  2  3  `4"C"_2 xx 6"C"_3 = (4xx3)/(2xx1) xx (6xx5xx4)/(3xx2xx1)` = 120 
1  4  `4"C"_1 xx 6"C"_4 = 4 xx 6"C"_2 = 4 xx (6xx5)/(2xx1) = 60` 

0  5  4C_{0} × 6C_{5} = 1 × 6C_{1} = 1 × 6 = 6  
Total number of ways = 186 
(ii) Almost two ladies are included means maximum of two ladies are included.
Possibilities  Ladies (4)  Gents (6)  Combinations 
1  2  3  `4"C"_2 xx 6"C"_3 = (4xx3)/(2xx1) xx (6xx5xx4)/(3xx2xx1)` = 120 
2  3  2  `4"C"_3 xx 6"C"_2 = (4xx3xx2)/(3xx2xx1) xx (6xx5)/(2xx1) = 60` 
3  4  1  4C_{4} × 6C_{1} = 1 × 6 = 6 
Total number of ways = 186 
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