Problem 1 :
A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of
(i) exactly 3 women?
(ii) at least 3 women?
(iii) at most 3 women?
Solution :
Number of men = 8
Number of women = 4
(i) exactly 3 women?
Number of ways = ^{4}C_{3} ⋅ ^{8}C_{4}
= 4 (70)
= 280
(ii) at least 3 women?
Number of ways = (^{4}C_{3} ⋅ ^{8}C_{4}) + (^{4}C_{4} ⋅ ^{8}C_{3})
= 4(70) + 1(56)
= 280 + 56
= 336
(iii) at most 3 women?
Number of ways
= (^{4}C_{0} ⋅ ^{8}C_{7}) + (^{4}C_{1} ⋅ ^{8}C_{6}) + (^{4}C_{2} ⋅ ^{8}C_{5}) + (^{4}C_{3} ⋅ ^{8}C_{4})
= 8 + 4(28) + 6(56) + 4(70)
= 8 + 112 + 336 + 280
= 736
Problem 2 :
7 relatives of a man comprises 4 ladies and 3 gentlemen, his wife also has 7 relatives; 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man’s relative and 3 of the wife’ s relatives?
Solution :
(i) 3 ladies from husband’s side and 3 gentlemen from wife’s side.
No. of ways in this case
= ^{4}C_{3} ⋅ ^{4}C_{3} = 4 ⋅ 4 = 16
(ii) 3 gentlemen from husband’s side and 3 ladies from wife’s side.
No. of ways in this case = ^{3}C_{3} ⋅ ^{3}C_{3} = 1 ⋅ 1 = 1
(iii) 2 ladies and one gentleman from husband’s side and lady and 2 gentlemen from wife’s side.
No. of ways in this case
= (^{4}C_{4} ⋅ ^{3}C_{1}) ⋅ (^{3}C_{1} ⋅ ^{4}C_{2}) = 6 ⋅ 3 ⋅ 3 ⋅ 6 = 324
(iv) One lady and 2 gentlemen from husband’s side and 2 ladies and one gentlemen from wife’s side.
No. of ways in this case
= (^{4}C_{1} ⋅ ^{3}C_{2}) ⋅ (^{3}C_{2} ⋅ ^{4}C_{1}) = 4 ⋅ 3 ⋅ 3 ⋅ 4 = 144
Hence the total no. of ways are
= 16 + 1 + 324 + 144 = 485 ways
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