Question

Find the differences between the greatest values in the following:

¹³C_r and ⁸C_r

Answer 1

The greatest value of ⁿC_r occurs at r = `"n"/2`, if n is even and at r = `("n" - 1)/2`, if n is odd.

The greatest value of ¹³C_r occurs at

r = `(13 - 1)/2` = 6    ...[∵ n = 13 is odd]

∴ the greatest value of ¹³C_r  

= ¹³C₆

= `(13!)/(6!7!)`

= `(13 xx 12 xx 11 xx 10 xx 9 xx 8)/(6 xx 5 xx 4 xx 3 xx 2 xx 1)`

= 13 × 11 × 2 × 3 × 2

= 1716

The greatest value of ⁸C_r occurs at

r = `8/2` = 4    ...[∵ n = 8 is even]

∴ the greatest value of ⁸C_r 

= ⁸C₄

= `(8!)/(4!4!)`

= `(8 xx 7 xx 6 xx 5)/(4 xx 3 xx 2 xx 1)`

= 7 × 2 × 5

= 70

∴ the difference between the greatest values of ¹³C_r and ⁸C_r

= 1716 – 70

= 1646