Mathematics

Problem

Let

P(x)=(1-5x)^{12}

. Answer the following questions based on the coefficients in this expansion. a. (i) Find the sum of all the coefficients of

P(x)

. b. (ii) Calculate the alternating sum of the coefficients of

P(x)

, that is,

\displaystyle\sum_{k=0}^{12}(-1)^k{12\brace k}(-5)^k

. c. (iii) Hence determine the sum of the coefficients of those terms in

P(x)

that contain an even power of

x

. Show all steps clearly.