01204211/homework5 counting 2

This is part of 01204211-58

Due: 19 Oct 2015

H.1

H.2 How many anagrams can you make from the word INNOVATION?

H.3 How many sorted lists of integer of length $k$ consisting only integers from $1$ to $n$ are there? (For example, when $k=2$ and $n=3$, there are 6 of them (i.e.: 1,1; 2,2; 3,3; 1,2; 1,3; and 2,3).

H.4 (LPV-3.4.1) In how many ways can you distribute all $n$ pennies to $k$ children if each child is supposed to get at least 2?

H.5 (LPV-3.6.3) Prove the following identity:

${\binom {n}{0}}{\binom {m}{k}}+{\binom {n}{1}}{\binom {m}{k-1}}+\cdots +{\binom {n}{k-1}}{\binom {m}{1}}+{\binom {n}{k}}{\binom {m}{0}}={\binom {n+m}{k}}.$

H.5 (LPV-3.8.8) Prove the following identity

$\sum _{k=0}^{n}{\binom {n}{k}}{\binom {k}{m}}={\binom {n}{m}}2^{n-m}.$