JEE MAIN - Mathematics (2021 - 25th February Morning Shift - No. 5)
The total number of positive integral solutions (x, y, z) such that xyz = 24 is :
36
24
45
30
Explanation
$$x.y.z = 24$$
$$x.y.z = {2^3}.\,{3^1}$$
Three 2 has to be distributed among x, y and z
Each may receive none, one or two
$$\therefore$$ Number of ways = $${}^{3 + 3 - 1}{C_{3 - 1}}$$ = $$^5{C_2}$$ ways
Similarly one 3 has to be distributed among x, y and z
$$ \therefore $$ Number of ways = $${}^{1 + 3 - 1}{C_{3 - 1}}$$ = $$^3{C_2}$$ ways
Total ways = $$^5{C_2}\,.{\,^3}{C_2}$$ = 30
$$x.y.z = {2^3}.\,{3^1}$$
Three 2 has to be distributed among x, y and z
Each may receive none, one or two
$$\therefore$$ Number of ways = $${}^{3 + 3 - 1}{C_{3 - 1}}$$ = $$^5{C_2}$$ ways
Similarly one 3 has to be distributed among x, y and z
$$ \therefore $$ Number of ways = $${}^{1 + 3 - 1}{C_{3 - 1}}$$ = $$^3{C_2}$$ ways
Total ways = $$^5{C_2}\,.{\,^3}{C_2}$$ = 30
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