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Alyssa Nardi

Alyssa N.

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Viewed Questions

(This becomes especially relevant in Chapter $6,$ though it makes an important point here.) In how many ways may we attach two identical red beads and two identical blue beads to the corners of a square (with one bead per corner) free to move around in (three-dimensional) space? (n)

Combinatorics Through Guided Discovery

Use the fact that $$ \frac{1-q^{2 i}}{1-q^{i}}=1+q^{i} $$ and the generating function for the number of partitions of an integer into distinct parts to show how the number of partitions of an integer $k$ into distinct parts is related to the number of partitions of an integer $k$ into odd parts. (h)

Combinatorics Through Guided Discovery

Use the method of partial fractions to convert the generating function of Problem 213 into the form $$ \frac{c}{x-r_{1}}+\frac{d}{x-r_{2}} $$ Use this to find a formula for $a_{n}$.

Combinatorics Through Guided Discovery

Suppose we start (at the end of month 0 ) with 10 pairs of baby rabbits, and that after baby rabbits mature for one month they begin to reproduce, with each pair producing two new pairs at the end of each month afterwards. Suppose further that over the time we observe the rabbits, none die. Let $a_{n}$ be the number of rabbits we have at the end of month $n$. Show that $a_{n}=a_{n-1}+2 a_{n-2}$. This is an example of a second order linear recurrence with constant coefficients. Using a method similar to that of Problem 211 , show that $$ \sum_{i=0}^{\infty} a_{i} x^{i}=\frac{10}{1-x-2 x^{2}} $$ This gives us the generating function for the sequence $a_{i}$ giving the population in month $i$; shortly we shall see a method for converting this to a solution to the recurrence.

Combinatorics Through Guided Discovery

Questions asked

INSTANT ANSWER

(a)Determine the generating function for the sequence {n^3}. (b)Determine the generating function for the sequence {1^3 + 2^3 +...+ n^3} Use (b) to prove that 1^3 + 2^3 +...+n^3 = [n^2(n+1)^2]/4

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ANSWERED

Keondre Parker verified

Numerade educator

What is the smallest positive integer m such that if m numbers are drawn at random from the set {1, 2, . . . , 100}, it is more likely than not that at least two of the numbers drawn are consecutive?

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INSTANT ANSWER

Let k ā‰„ 3 be an integer. Determine S(k, k āˆ’ 2) and S(k, 3). Justify your answers.

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INSTANT ANSWER

Problem 142. (Relevant in Appendix C.) The total number of partitions of a k-element set is denoted by B(k) and is called the k-th Bell number. Thus B(1) = 1 and B(2) = 2. (a) Show, by explicitly exhibiting the partitions, that B(3) = 5. (b) Find a recurrence that expresses B(k) in terms of B(n) for n < k and prove your formula correct in as many ways as you can. (h) (c) FindB(k)fork=4,5,6.

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ANSWERED

Nick Johnson verified

Numerade educator

How many integer solutions are there to the equation x + y + z = 20 with x ā‰„ āˆ’2, y > 2, and z ā‰„ 0?

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INSTANT ANSWER

let n be a positive integer. prove that the sumation of n,j=0 (-1)^j (n choose j)^2 = 0 if n is odd and (-1)^m(2m choose m) if n =2m

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ANSWERED

Sam Stansfield verified

Numerade educator

Let n be a multiple of 8. Show that there are 2^(nāˆ’2) subsets of an n-set with sizes congruent to 3 (mod 4)

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