Question

The total number of positive integral solutions (x, y, z) such that xyz = 24 is :

(1) 36

(2) 24

(3) 45

(4) 30

Answer 1

Correct option is (4) 30

The number 24 can be factored as: 24 = 2³ × 3.

Hence, you need x, y, z to be of the form  2^a3^b You have 3 powers of 2 and one power of 3 which you need to distribute between three numbers.

For 2′s, this can be done in ⁵C₂ ​= 10 ways (a₁ ​+ a₂​ + a_3​ = 3 ⇒ ^n+r−1C_r−1​), and for 3 in 3 ways(b_{1 ​}+ b₂​ + b₃ ​= 1).

Hence, the answer is 3 × 10 = 30.

Answer 2

Correct option is (4) 30

No. of non-negative intergal sol = ⁵C₂ = 10  &  β₁+β₂+β₃ = 1

No. of non-negative intergal solⁿ = ³C₂ = 3

Total ways = 10 x 3 = 30.