Question

Which is larger : C_{93, 30} or C_{93, 62} ? Justify your answer.

Answer 1

Please note ⁿC_r = n! / [(n-r)!*r!].

Also ⁿC_r = ⁿC_n-r eg ⁵C₂ = ⁵C₃

C93,30 = ⁹³C₃₀ = ⁹³C_93-30 = ⁹³C₆₃

C93,62 = ⁹³C₆₂

As we go on from ⁹³C₁ till the middle values 93 which are the 46th and 47th values, the values will keep on increasing. Like ⁹³C₀ = 1, ⁹³C₁ = 93, ⁹³C₂ = 4278 and so on.

The 46th and the 47th values will be equal as ⁿC_r = ⁿCn-r, so ⁹³C₄₆ = ⁹³C_{97 - 46} = ⁹³C₄₇.

Then after than as r increases, ⁿCr, keeps on reducing until we come to ⁹³C₉₃ = 1.

Since 63 is > 62 and is after the 47th value, therefore ⁹³C₆₂ will be larger than ⁹³C₆₃.