Answer 1

Use `""^"n""C"_r` has maximum value if,

(a) r = `"n"/2` when n is even

(b) r = `("n" - 1)/2  "or" ("n" + 1)/2` when n is odd

`""^13"C"_r  "and"  ""^8"C"_r`

Maximum value of `""^13"C"_r  "occurs at"  r = 12/2 = 6  "or  r = 14/2` = 7 

∴ Maximum value of `""^13"C"_r = ""13"C"_6`

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

= 1716

Maximum value of `""^8"C"_r  "occurs at"  r = 8/2` = 6

∴ Maximum value of `""^8"C"_r = ""^8"C"_4`

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

= 70

∴ The difference between the maximum values of `""^13"C"_r  "and"  ""^8"C"_r`

= 1716 – 70
= 16468.