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# MTH 4150: Midterm 1 Practice 2

Problem 1

Provide an example of a function $$f:\{ A,B,C,D,E \}\to \{ 1,2,A,B,C,D \}$$ that satisfies $$f(B)=1$$ and that is not one-to-one.

Problem 2

A sequence of integers $$\left(a_n\right)_{n=1}^{\infty}$$ satisfies $$a_1=6$$, $$a_2=20$$, and $$a_{n+2}=2a_{n+1}+8a_n$$ for $$n\geq 1$$. Prove that $$\displaystyle a_n=\frac{4 \cdot 4^n-{ } \left(-2\right)^n }{3}$$ for every $$n\geq 1$$.

Problem 3

In how many ways can one distribute $$26$$ red and $$30$$ green apples to $$21$$ distinguishable people so that each person receives at least one apple of each color?

Red apples are indistinguishable among themselves, and green apples are indistinguishable among themselves.

Problem 4

• (a) Provide an example of a bijection $$f:\{1,2,\dots, 28 \}\to \{ 1,2,\dots, 28 \}$$ such that $$f(1)\neq 1$$ but $$f(f(1))=1$$.

• (b) Determine the number of bijections $$f:\{1,2,\dots, 28 \}\to \{ 1,2,\dots, 28 \}$$ such that $$f(f(1))=1$$.

Problem 5

A sequence whose all terms are from $$\{ A,B,C,D, 1,2\}$$ is called silver if no two consecutive terms are letters. For example, the sequences $$(A,1,1,B,1)$$ and $$(1,A,1,B,1)$$ are silver, while $$(1,A,B,1,1)$$ and $$(1,A,A,1,B)$$ are not. Determine the formula for the number of silver sequences of length $$n$$.

Problem 6

Assume that $$a_1$$, $$\dots$$, $$a_{20}$$ are positive integers such that $$0< a_1< a_2< \cdots< a_{20}< 100$$. Prove that there are 4 numbers $$i$$, $$j$$, $$k$$, and $$l$$ such that $$1\leq i< j< k< l\leq 20$$ and $$a_j-a_i=a_l-a_k$$.