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Question
Find the differences between the largest values in the following: `""^13"C"_r "and" ""^8"C"_r`
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Solution
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` = 4
∴ 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
= 1646.
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