Question

From $7$ Englishmen and $4$ Americans a committee of $6$ is to be formed; in how many ways can this be done. (1) when the committee contains exactly $2$ Americans. (2) at least $2$ Americans.

Answer 1

(1) We have to choose $2$ Americans and $4$ Englishmen.

The number of ways in which the Americans can be chosen is ${}^4{C}_2$4C2; and the number of ways in which the Englishmen can be chosen is ${}^7{C}_4$. Each of the required number of ways

${=}^4{C}_2{\times }^7{C}_4$

$=210$.

(2) The committee may contain $2,3$ or $4$ Americans.

We shall exhaust all the suitable combinations by forming all the groups containing $2$ Americans and $4$ Englishmen; then $3$ Americans and $3$ Englishmen; and lastly $4$ Americans and $2$ Englishmen.

The sum of the three results will give the answer. Hence the required number of ways

$={}^4{C}_2\times {}^7{C}_4+{}^4{C}_3\times {}^7{C}_3+{}^4{C}_4\times {}^7{C}_2$

$=210+140+41=371$