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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 \( a_j-a_i=a_l-a_k \).