A committee of 8 is to be formed from 16 men and 10 women. In how many ways can the committee be formed if i) there are no restrictions. ii) there must be 4 men and 4 women. iii) there should be an even number of women. iv) there are more women than men. v) there are atleast 6 men.

Solution:

Given, 16 men and 10 women.

i) No. of ways when there are no restrictions

$=\ ^{26}C_8 \\=1562275$

ii) No. of ways when there must be 4 men and 4 women

$=\ ^{16}C_4\times \ ^{10}C_4 \\=382200$

iii) No. of ways when there should be an even number of women

$=\ ^{16}C_6\times \ ^{10}C_2+\ ^{16}C_4\times \ ^{10}C_4+\ ^{16}C_2\times \ ^{10}C_6+\ ^{16}C_0\times \ ^{10}C_8 \\=8008\times45+1820\times210+120\times210+1\times45 \\=767805$

iv) No. of ways when there are more women than men

$=\ ^{16}C_3\times \ ^{10}C_5+\ ^{16}C_2\times \ ^{10}C_6+\ ^{16}C_1\times \ ^{10}C_7+\ ^{16}C_0\times \ ^{10}C_8 \\=560\times 252+120\times210+16\times120+1\times45 \\=168285$

v) No. of ways when there are at least 6 men

$=\ ^{16}C_6\times \ ^{10}C_2+\ ^{16}C_7\times \ ^{10}C_1+\ ^{16}C_8\times \ ^{10}C_0 \\=8008\times45+11440\times10+12870\times1 \\=487630$