Use the binomial formula to the coefficient of the #t^22p^3# term in the expression #(t+2p)^25#?

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Answer 1 · 2018-04-27T11:50:18

Coefficient of #t^22p^3# is #2300#.

In the expansion of #(a+b)^n#, binomial formula gives the #(r+1)^(th)# term as #C_r^na^(n-r)b^r#

Therefore in #(t+2p)^25#, #(r+1)^(th)# term is #C_r^25t^(25-r)(2p)^r#

i.e. #2^rC_r^25t^(25-r)p^r#

As we are looking for term containing #t^22p^3#, our #r=3#

and #(3+1)^(th)# term containing #t^22p^3# is

#2^3*C_3^25t^22p^3#

= #8*(25*24*23)/(1*2*3)t^22p^3#

= #8*2300t^22p^3#

= #18400t^22p^3#

Hence coefficient of #t^22p^3# is #2300#.