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प्रश्न
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.
उत्तर
(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 | 4C0 × 6C5 = 1 × 6C1 = 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 | 4C4 × 6C1 = 1 × 6 = 6 |
| Total number of ways = 186 |
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