3. Recall that a composition of a positive integer n is a way of writing n as a sum of positive integers, called parts, which may appear in any order. It turns out to be interesting to count the number of compositions of n using only odd parts. Here is a table for small values of n: n 1 2 3 4 5 compositions of n with only odd parts 1 1+1 3, 1+1+1 3+1, 1+3, 1+1+1+1 5, 3+1+1, 1+3+1, 1+1+3, 1+1+1+1+1 In class we showed that each composition of n comes from a composition of n - 1 by doing one of two things: 2 1. adding 1 as a new last part 2. adding 1 to the current last part The first operation still gets us from a composition of n-1 with all parts odd to a composition of n with all parts odd. The second operation fails, but there is a replacement: add 2 to the current last part of a composition of n - 2 with all parts odd. Thus, for example, the compositions of 5 above with last part 1 come from adding 1 as a new last part to the compositions of 4 above. The compositions of 5 above whose last part is not 1 come from adding 2 to the last part of the compositions of 3 above. Recall that the Fibonacci numbers are defined by Fn+1 = Fn + Fn-1 for n ≥ 1, with Fo = 0 and F₁ = 1. Prove by induction that the number of compositions of n with all parts odd is Fn.

A basket contains 2 pink, 3 yellow, 8 red, and 1 black jelly beans. You grab a handful of 4 jelly beans. (a) How many groups of 4 jelly beans include exactly 1 of each color?   (b) How many groups of 4 jelly beans include at most 1 yellow?

Select the number line that shows that two opposite numbers have a sum of 0. A. B. C. D. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 6 4 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 2 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 3 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 SUBMIT

2. I found some bingo cards in our classroom. A bingo card seems to have 5 rows and 5 columns. The center cell is blank, but the remaining cells each contain a number. No two cells contain the same number. In the first column, integers from 1 to 15 are used. In the 2nd column, integers from 16 to 30 are used, and so on. Thus, in the 5th column, integers from 61 to 75 are used. How many different bingo cards are there? Explain.

a)Select a variety of one- and two digit numbers between 1 and 99 and multiply by 9. Record your results. b) Find the sum of the digits in each of your products in part (a). If the sum is not a one digit number find the sum of the digits of the resulting sum again until you obtain a one -digit number, c) make a conjecture about the sum of digits when a one- or two digit number is multiplied by 9