A group has 8 men and 7 women. In how many ways can a committee of 5 people be formed if the number of women is at least 3?

Question: A group has 8 men and 7 women. In how many ways can a committee of 5 people be formed if the number of women is at least 3?
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​Solution:
​Given that,
​Total committee size = 5

And for women ≥ 3, possible distributions:
Three ways to formed the committee
1. 3 women + 2 men
2. 4 women + 1 man
3. 5 women + 0 men
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​Now, 1st case- 3 women + 2 men
​Choose 3 women from 7, (⁷C₃) = 35
​Choose 2 men from 8, (⁸C₂) = 28
​∴ Total ways = 35 × 28 = 980
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​2nd case- 4 women + 1 man
​Choose 4 women from 7, (⁷C₄) = 35
Choose 1 man from 8, (⁸C₁) = 8
​∴ Total ways = 35 × 8 = 280
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​And 3rd case-​5 women + 0 men
Choose 5 women from 7, (⁷C₅) = 21
No men to choose
​∴ Total ways = 21
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​∴ ​Total ways = 980 + 280 + 21= 1281