First slide

Combinations

Question

If a, b, c are three natural numbers in A.P. such that a + b + c = 21, then the possible number of values of a, b, c is

Moderate

A

13
B

14
C

15
D

16

Solution

Let b = a + d and c = a + 2d.
Given: a + a + d + a + 2d = 21 ⇒ a + d = 7
∴ a + c = 14 and b = 7
Now, the number of positive integral solutions of a + c = 14 is equal to coefficient of x¹⁴ in (x + x² + x³ +….)² = Coefficient of x¹² in (1 + x + x² …)² = ¹³C₁₂ = 13.

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