Question

Find the differences between the greatest values in the following:

¹⁵C_r and ¹¹C_r 

Answer 1

Greatest value of ¹⁵C_r

Here n = 15, which is odd

Greatest value of ⁿC_r occurs at r = `("n" - 1)/2` if n is odd

∴ r = `("n" - 1)/2`

∴ r = `(15 - 1)/2` = 7

∴ Greatest value of ¹⁵C_r

= ¹⁵C₇

= `(15!)/(7!8!)`

= `(15 xx 14 xx 13 xx 12 xx 11 xx 10 xx 9 xx 8!)/(7 xx 6 xx 5 xx 4 xx 3 xx 2 xx 1 xx 8!)`

= 6435

Also, for greatest value of ¹¹C_r

n = 11, which is odd

∴ r = `(11 - 1)/2` = 5

∴ ¹¹C₅ = `(11!)/(5!6!)`

= `(11 xx 10 xx 9 xx 8 xx 7 xx 6!)/(5 xx 4 xx 3 xx 2 xx 1 xx 6!)`

= 462

∴ Difference between the greatest values of ¹⁵C_r and ¹¹C_r 

= ¹⁵C_r – ¹¹C_r

= 6435 – 462

= 5973