First slide

Combinations

Question

A is a set containing n elements. A subset P of A is chosen. The set A is reconstructed by replacing the elements of P. A subset Q of A is again chosen. The number of ways of choosing P and Q so that $P \cap Q$ contains exactly two elements is

Difficult

A

$9^{n} C_{2}$
B

$3^{n} -^{n} C_{2}$
C

$^{n} C_{2} 3^{n - 2}$
D

$4^{n} - 3^{n}$

Solution

Let $A = {a_{1}, a_{2}, a_{3}, ........... a_{n}} .$ For any $a_{i} \in A,$ we may have following situations.
$(i) a_{i} \in P, a_{i} \in Q$

$(i i) a_{i} \in P, a_{i} \notin Q$

$(i i i) a_{i} \notin P, a_{i} \in Q$

$(i i i) a_{i} \notin P, a_{i} \notin Q$

$∴ P \cap Q$ contains exactly two elements. Taking 2 elements belonging to case $(i) a n d (n - 2)$ elements will belong to case $(i i) o r (i i i) o r (i v)$

$∴$ Number of ways $=^{n} C_{2} \times 3^{n - 2} .$

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