IPMAT Indore 2023 - QA (SA) Q3 Explanation

Explanation:

(1160)

We have to find number of positive integral solutions of 21 ≤ a + b + c ≤ 25 or, we can say, we have to find number of positive integral solution of the following five equations,

a + b + c = 21,

a + b + c = 22,

a + b + c = 23,

a + b + c = 24, and

a + b + c = 25.

We know that, number of positive integral solutions of the equation A+B+C = n (a constant) is given by n-1Cr-1, where r = number of variables (which is 3 in this case)

∴ The equation 𝑎 +𝑏 +𝑐

= 21 will have 21-1C3-1 = 20C2

Similarly, for other 4 equation, we will have number of positive integral solutions as

21C2, 22C2, 23C2 and 24C2.

Adding all 20C2 + 21C2 + 22C2 + 23C2 and 24C2

We get (20 * 19)/2 + (21 * 20)/2 + (22 * 4)/2 + (22 * 22)/2 + (24 * 23)/2

= 1/2 * [380 + 420 + 462 + 506 + 552]

= 1/2 * [2320] = 1160 Ans .