Question

please do both 4 and 5 and show work! will upvote fast

Problem VII.4. How many non-negative integer vectors are there such that where exactly k of the entries zero? N Problem VII.5. Prove the following: () ik

Answer 1

R-k Qar - 2 N where exactly k are gerees do, we have { di = N here the number of solution is same as coefficient of an in the expansion of [1 + x + x² + ... kok which is equal to R-k R- ktN-te Now we have ex no. of ways = choices for o's, hence total d ex. R-ktN-CN-

(1:1) First we write it as N-K + pk-1+1 / K-145 K-1 K-U jak- Lang - (9) EQ - (+) + * "+ Ž ) Nove since (K-1) = 1 = (4) (43") +- - *43-(43") Hence RHS = (kt) + 2 kt) - (Kid) + 3 (*-* * * *' ) - N-K N-X K+2 ( kiti K-1 13 K-1+ N-KH I K-1+1 Slence Les = RHS