Question

If ⁸C_r – ⁷C₃ = ⁷C₂, find r.

Answer 1

To find r, 

Lets consider the given expression,

⁸C_r – ⁷C₃ = ⁷C₂

⁸C_r = ⁷C₂ + ⁷C₃

As we know that ⁿC_r + ⁿC_{r + 1} = ^{n + 1}C_{r + 1}

⁸C_r = ^{7 + 1}C_{2 + 1}

⁸C_r = ⁸C₃

Then, we know that if ⁿC_p = ⁿC_q, now one of the following conditions need to be satisfied:

(i) p = q

(ii) n = p + q

Therefore from the question ⁸C_r = ⁸C₃, we can say that

Lets check for condition (i)

r = 3

Lets also check for condition (ii)

8 = 3 + r

r = 5

Hence, the values of ‘r’ are 3 and 5.