#Unit 3-Lecture 4：Repetitions & Binomial Theorem

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A k-to-1 function maps exactly k elements of the domain to every element of the codomain.
Rule:

I f f : A \to B i s k - t o - l, t h e n | A | = k \cdot | B |

Subset Rule:
The number of k-element subsets of an n-element set is:

(n k) = n ! k ! ( n - k ) !

Definition：
For n,k1,...,km∈N, such that k1+k2+...+km=n, define the mutinomial coefficient

(n k 1 , k 2 , . . . , k m) : : = n ! k 1 ! k 2 ! . . . k m !

Subset Split Rule:
The number of

(k1,k2,...,km) -splits of an n-elements set is:

(n k 1 , k 2 , . . . , k m)

Bookkeeper Rule:
Let

l1,...,lm be distinct elements. The number of sequences with

k1 occurences of

l1, and

k2 occurence of

l2… and

km occurence of

lm is:

(k 1 + k 2 + . . . + k m k 1 , k 2 , . . . , k m)

The Binomial Theorem:
For all

n∈N and

a,b∈R：

(a + b) n = \sum k = 0 n (n k) a n - k b k

The multinomial Theorem:
For all

n∈N:

(z 1 + z 2 + . . . + z m) n = \sum k 1 + k 2 + . . . + k m = n (n k 1 , k 2 , . . . , k m) z k 1 1 z k 2 2 . . . z k m m

[1] Lehman E, Leighton F H, Meyer A R. Mathematics for Computer Science[J]. 2015.

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