Question

A firm of Chartered Accountants in Bombay has to send 10 clerks to 5 different companies, two clerks in each. Two of the companies are in Bombay and the others are outside. Two of the clerks prefer to work in Bombay while three others prefer to work outside. In how many ways can the assignment be made if thepreferences are to be satisfied

Answer 1

$2$ clerks who prefer Bombay are to be sent to $2$ different companies in Bombay,
and Out of remaining $5$ clerks (excluding $3$ clerks who prefer for outside) $2$ clerks are chosen in ${}^5{C}_2$ ways.

Now these 4 can be sent to 2 different companies into 2 groups of 2 each in ${}^4{C}_2$ ways

$\Rightarrow {}^5{C}_2\cdot {}^4{C}_2$

Now for outside companies we have 6 clerks remaining we select them as (2 for each company) ${=}^6{C}_2\times {}^4{C}_2\times {}^2{C}_2$

Desired ways $={(}^5{C}_2\times {}^4{C}_2){(}^6{C}_2\times {}^4{C}_2\times {}^2{C}_2)=5400$ ways.