Question

Find the differences between the largest values in the following: `""^15"C"_r  "and"  ""^11"C"_r`

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

`""^15"C"_r  "and"  ""^11"C"_r`

Maximum value of `""^15"C"_r  "occurs at"  r = 14/2 = 7  "or  r = 16/2` = 8 

∴ Maximum value of `""^15"C"_r = ""^15"C"_7 = ""^15"C"_8`

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

= 6435

Maximum value of `""^11"C"_r  "occurs at"  r = 10/2 = 5  "or" r = 12/2` = 6

∴ Maximum value of `""^11"C"_r = ""^11"C"_5 = ""^11"C"_6`

`""^11"C"_r = ""^11"C"_5 = (11 xx 10 xx 9 xx 8 xx 7)/(5 xx 4 xx 3 xx 2 xx 1)`

= 462

∴ The difference between the maximum values of `""^15"C"_r  "and"  ""^11"C"_r`

= 6435 – 462
= 5973.