Question

Find the differences between the largest values in the following: `""^14"C"_r  "and"  ""^12"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

`""^14"C"_r  "and"  ""^12"C"_r`

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

∴ Maximum value of `""^14"C"_r = ""^14"C"_7`

= `(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)`

= 3432

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

∴ Maximum value of `""^12"C"_r = ""^12"C"_6`

= `(12 xx 11 xx 10 xx 9 xx 8 xx 7)/(8 xx 7 xx 6 xx 5 xx 4 xx 3 xx 2)`

= 924

∴ The difference between the maximum values of `""^14"C"_r  "and"  ""^12"C"_r`

= 3432 – 924
= 2508.