The total number of ways of forming a committ | vedclass

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(D) We need to form a committee of $5$ members from $7$ Indians $(I)$,$6$ Americans $(A)$,$5$ Russians $(R)$,and $4$ Australians $(AU)$ such that each

Vedclass · Accepted Answer

$7560$

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(D) We need to form a committee of $5$ members from $7$ Indians $(I)$,$6$ Americans $(A)$,$5$ Russians $(R)$,and $4$ Australians $(AU)$ such that each country is represented at least once.Since the total number of members is $5$ and there are $4$ countries,one country must have $2$ representatives,and the other three countries must have $1$ representative each.The possible distributions are:$1$. $2I, 1A, 1R, 1AU: {^7C_2} 	imes {^6C_1} 	imes {^5C_1} 	imes {^4C_1} = 21 	imes 6 	imes 5 	imes 4 = 2520$$2$. $1I, 2A, 1R, 1AU: {^7C_1} 	imes {^6C_2} 	imes {^5C_1} 	imes {^4C_1} = 7 	imes 15 	imes 5 	imes 4 = 2100$$3$. $1I, 1A, 2R, 1AU: {^7C_1} 	imes {^6C_1} 	imes {^5C_2} 	imes {^4C_1} = 7 	imes 6 	imes 10 	imes 4 = 1680$$4$. $1I, 1A, 1R, 2AU: {^7C_1} 	imes {^6C_1} 	imes {^5C_1} 	imes {^4C_2} = 7 	imes 6 	imes 5 	imes 6 = 1260$Total ways $= 2520 + 2100 + 1680 + 1260 = 7560$.

The total number of ways of forming a committee of $5$ members out of $7$ Indians,$6$ Americans,$5$ Russians,and $4$ Australians such that every committee contains at least one member from each country is:

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