Harvard University Combinatorics Compositions and Collatz conjecture Questions

I need Help in Combinatorics Class Homework. It is only six questions and need to show work. It should be typed on ms word. I can extend the time up to 12 hours only if it is needed. Thank you so much for your help.

Instruction: Show complete solution and answer all the problems.
1. (a) Find the number of (strong) compositions of 5. Find the number of (strong) compositions of
10 into all even parts.
(b) Based on your observations in part (a), state and prove a conjecture about the number of
(strong) compositions of n into even parts.
2. Prove that for 𝑛 ≥ 2 , the number of partitions of 𝑛 in which the two largest parts are equal is
given by 𝑝(𝑛) − 𝑝(𝑛 − 1).
3. In discussion, we proved that the number of partitions of 𝑛 into at most 𝑘 parts equals the
number of partitions of 𝑛 into parts of size at most 𝑘. Use this to find an expression for the number
of partitions of 𝑛 into precisely 𝑘 parts.
4. A partition is self-conjugate if it equals its conjugate. Prove the number of self-conjugate partitions
of 𝑛 has the same parity as 𝑝(𝑛) (that is, the number of self-conjugate partitions of 𝑛 is even if and
only if 𝑝(𝑛) is even).
5.Find an expression for the number of partitions of 𝑛 in which each part appears an even number
of times (your expression is allowed to use 𝑝(𝑘 ) for 𝑘 ≤ 𝑛 ). For example, this includes the partition
4 + 4 + 3 + 3 + 3 + 3 + 1 + 1 , but not 8 + 3 + 3 + 1 + 1 .
6. Prove that the number of partitions of 𝑛 into odd parts equals the number of partitions of n into
distinct parts.