Question

Find the differences between the greatest values in the following:

¹⁴C_r and ¹²C_r 

Answer 1

Greatest value of ¹⁴C_r

Here n = 14, which is even

Greatest value of ⁿC_r occurs at r = `"n"/2` if n is even

∴ r = `"n"/2`

∴ r = `14/2` = 7

∴ Greatest value of ¹⁴C_r = ¹⁴C₇ = `(14!)/(7!7!)` 

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

= 3432

Also, for greatest value of ¹²C_r

n = 12, which is even

∴ r = `12/2` = 6

∴ ¹²C₆ = `(12!)/(6!6!)`

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

= 924

∴ Difference between the greatest values of ¹⁴C_r and ¹²C_r  

= ¹⁴C_r and ¹²C_r 

= 3432 – 924

= 2508