In: Advanced Math

*Combinatorics*

Prove bell number B(n)<n!

so bell number of a number of was in which that number can be partitioned. above result is true for N.2

let S(p,k) be the number in which p numbers can be put into k sets.

S(p+1,k)=kS(p,k)+S(p,k-1)

so this is because there are two positibilities for upcoming number first Add as a singleton set (in this the partition of p numbers in k-1 set gives away a new posibility for p+1 numbers into k set).

another it goes into a previously defined set (going into a set give rise to S(n,k) posibilities, as there are k sets so kS(p,k) poibilities. now B(n)=

now use induction for n=1 B(3)=5 <3!

suppose it is true for p=n

we will prove for p=n+1

B(n+1)=

rewriting above as S(n,n+1)=0 and S(n,-1)=0

as hypothesis B(n)<n! then (n+1)B(n) is atleast n+1 less than (n+1)! so above is true.

If n>=2, prove the number of prime factors of n is less than
2ln n.

Prove that for all integers n ≥ 2, the number p(n) − p(n − 1) is
equal
to the number of partitions of n in which the two largest parts
are
equal.

let n belongs to N and let a, b belong to Z. prove that a is
congruent to b, mod n, if and only if a and b have the same
remainder when divided by n.

Prove by induction on n that the number of distinct handshakes
between n ≥ 2 people in a room is n*(n − 1)/2 .
Remember to state the inductive hypothesis!

Using the pumping lemma, prove that the language {1^n | n is a
prime number} is not regular.

Prove that the number of partitions of n into parts of size 1
and 2 is equal to the number of partitions of n + 3 into exactly
two distinct parts

This is a Combinatorics question.
Find a generating function for a sub r, the number of
ways:
(1) To distribute ridentical objects into seven
distinct boxes with an odd numbet of objects not exceeding nine in
the first three boxes and between four and ten in the other
boxes.

This is a Combinatorics Problem
Consider the problem of finding the number of ways to distribute
7 identical pieces of candy to 3 children so that no
child gets more than 4 pieces. Except Stanley (one of the 3
children) has had too much candy already, so he’s only
allowed up to 2 pieces. Write a generating function & use your
generating function to solve this problem.

This is a combinatorics problem
Suppose we wish to find the number of integer solutions to the
equation below, where 3 ≤ x1 ≤ 9, 0 ≤ x2 ≤ 8, and
7 ≤ x3 ≤ 17.
x1 + x2 + x3 = r
Write a generating function for this problem, and use it to
solve this problem for r = 20.

Prove the language of strings over {a, b} of the form (b^m)(a^n)
, 0 ≤ m < n-2 isn’t regular.
(I'm using the ^ notation but your free to make yours
bman instead of (b^m)(a^n) )
Use the pumping lemma for regular languages.

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