Let A be a set of n(≥ 3) distinct elements. The number of triplets (x, y, z) of the elements of A in which at least two coordinates are equal to

Question

Let A be a set of n(≥ 3) distinct elements. The number of triplets (x, y, z) of the elements of A in which at least two coordinates are equal to (A)  n P3 (B) n3- n P3 (C) 3 n2-2 n (D) 3 n2(n-1)

Tardigrade Admin · Accepted Answer

Total number of triplets without restriction =n × n × n . The number of triplets with all different coordinates = n P3 ∴ The required number of triplets =n3-n(n-1)(n-2)

Q. Let $A$ be a set of $n (\geq 3)$ distinct elements. The number of triplets $(x, y, z)$ of the elements of $A$ in which at least two coordinates are equal to

$^{n} P_{3}$

$n^{3} -^{n} P_{3}$

$3 n^{2} - 2 n$

$3 n^{2} (n - 1)$

Total number of triplets without restriction $= n \times n \times n .$
The number of triplets with all different coordinates $=^{n} P_{3}$
$∴$ The required number of triplets $= n^{3} - n (n - 1) (n - 2)$

Q. Let A be a set of n(≥3) distinct elements. The number of triplets (x,y,z) of the elements of A in which at least two coordinates are equal to

nP3​

n3−nP3​

3n2−2n

3n2(n−1)