NDA 1 2024 Maths, question 5

Topic: Algebra

Q5

A man has $7$ relatives ( $4$ women and $3$ men). His wife also has $7$ relatives ( $3$ women and $4$ men). In how many ways can they invite $3$ women and $3$ men so that $3$ of them are man's relatives and $3$ of them are his wife's relatives ?

(a) $340$
(b) $484$
(c) $485$ Correct answer
(d) $469$

Solution.

Answer: (c) $485$

Let k of the man's three invited relatives be women.
Then choose k from his 4 women and 3−k from his 3 men; choose 3−k from his wife's 3 women and k from her 4 men.
Summing C(4,k)²C(3,3−k)² for k=0,1,2,3 gives 1+144+324+16=485, treating the two groups of relatives as distinct.