Chapter 9.1, Problem 94E

DISCUSS • DISCOVER: Why ls (;) the Same as C(n,r)? This explains why binomial coefficients (;) that appear in expansion of (x + y)" are the same as C(n, r). num-bar Of ways of choosing r objects from n objects. first. Note that expanding a binomial using only the Distributive property gives
   ${(x+y)}^2=(x+y)(x+y)$
$=(x+y)x+(x+y)y$
$=xx+xy+yx+yy$
${(x+y)}^3=(x+y)x+(xx+xy+yx+yy)$
$=xxx+xxy+xyx+xyy+yxx$
$+yxy+yyx+yyy$

(a) Expand ${(x+y)}^5$ using only the Distributive Property.
(b) Write all that the term represents $x^2{y}^3$ . These are all the terms that contain two x' s and three y's.
(C) Note that tie two x's appear in all possible positions. Conclude that the number of terms that represent $x^2{y}^3$ is C (5,2)
(d) In general, explain why (;) in the Binomial theorem is the same as C(n, r).