Homework 8 Quick Answers

1. Rewrite the generic term as C(m,m-k)*C(n,r+k). In this form the target, C(m+r,m+n) accounts for all paths to location m+r at depth m+n. The lead term, C(m,m)*C(n,r) says do m of the m+r right terms at level m, then drop down the remaining n levels via all paths with r right turns; the next term C(m,m-1)*C(n,r+1), says do one fewer right turn up to depth m, then add one extra in the second half drop to level m+n. Every path through level m is accounted for in this way, each with the appropriate ending paths to get to C(m+r,m+n).

2. Diferentiate (1-x)n and its binomial expansion (then set x = 1).

3. Rewrite general term as C(n,n-k)*C(m,k). Given a set of m+n things, count size n subsets, beginning with all the n things, then one n thing missing, + one chosen m thing, then 2 n things missing, + 2 chosen m things, and so forth. So: C(m+n,n).

4. C(23,15)- C(9,1)*C(18,10)+C(9,2)*C(13,5)-C(9,3)

5. Coefficient of xr in (1+x+x2+....+xk +...)5

6. Fourth expression captures a; third expression captures b-a; and so forth.

7. This is x35*(1+x+x2...)7; coefficient of x(r-35) is C(r-41,r-35).

8. (1+x+x2..+x9)5, coefficient of xr

9. Treat as 10 balls in 5 boxes, each box gets 0 - 5 balls. Seek coefficient of x10 in (1+x+x2..+x5)5

10. (1+x+x2+x3)3(1+x+x2+....+xk +...)2 seek coefficient of x10.

11. Think of this as 1/(1-u), where u is (x+x2+....+x6)

-- RobbieMoll - 2012-05-01

Topic revision: r3 - 2012-05-03 - RobbieMoll
 
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