Question

Evaluate the expression (y+1)^4 – (y-1)^4.

(a) 3y^2 + 2y^5

(b) 7(y^4 + y^2 + y)

(c) 8(y^3 + y^1)

(d) y + y^2 + y^3

The question was posed to me in an interview for internship.

The origin of the question is Counting topic in portion Counting of Discrete Mathematics

Answer 1

Right option is (c) 8(y^3 + y^1)

The best explanation: By using Binomial theorem,the expression (y+1)^4 – (y-1)^4 can be expanded as = (y+1)^4 = ^4C0y^4 + ^4C1y^3 + ^4C2y^2 + ^4C3y^1 + ^4C4y^0 and (y-1)^4 = ^4C0y^4 – ^4C1y^3 + ^4C2y^2 – ^4C3y^1 + ^4C4y^0. Now, (y+1)^4 – (y-1)^4 = (^4C0y^4 + ^4C1y^3 + ^4C2y^2 + ^4C3y^1 + ^4C4y^0)  – (^4C0y^4 – ^4C1y^3 + ^4C2y^2 – ^4C3y^1 + ^4C4y^0) = 2(^4C1y^3 + ^4C3y^1) = 8(y^3 + y^1).