Let $$A$$ be a set of $$n\left( { \geqslant 3} \right)$$ distinct elements. The number of triplets $$(x, y, z)$$ of the elements of $$A$$ in which at least two coordinates are equal is
A.
$$^n{P_3}$$
B.
$${n^3}{ - ^n}{P_3}$$
C.
$$3{n^2} - 2n$$
D.
$$3{n^2}\left( {n - 1} \right)$$
Answer :
$$3{n^2} - 2n$$
Solution :
Total number of triplets without restriction $$ = n \times n \times n.$$
The number of triplets with all different co-ordinates $$ = {\,^n}{P_3}.$$
∴ the required number of triplets $$ = {n^3} - n\left( {n - 1} \right)\left( {n - 2} \right).$$
Releted MCQ Question on Algebra >> Permutation and Combination
Releted Question 1
$$^n{C_{r - 1}} = 36,{\,^n}{C_r} = 84$$ and $$^n{C_{r + 1}} = 126,$$ then $$r$$ is:
Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated are
Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from amongst the chairs marked 1 to 4 ; and then the men select the chairs from amongst the remaining. The number of possible arrangements is