Question

Question #3 please with some explanations. Thank you!

E - 21% O Tue 10:16 PM Q 2017.pdf (page 1 of 3) @ Q Search • Know what the symbol [x4] P(x) means. Exercises A. Handout Questions In the questions below 'closed form' means writing the generating function not as an infinite series using "...", but writing it in terms of rational functions. For example, I is the closed form of 1+2++ ... 1. For each equation, express the number of solutions as the coefficient of a polynomial. (a) bi + b2 + b3 = 14 where 61, 62, 63 > 0 and b1,b2 are odd while b3 is even. (b) b + b + b3 + bx = 17 where 2 < b < 5, 62, 63, 64 > 0, b2 is even, bz is odd, and be is a multiple of 3. 2. (a) Consider the number of ways 24 marbles can be distributed among 4 distinct containers so that each container receives at least three marbles. Give the answer as a coefficient of a generating function (written in closed form). (b) Redo part (a) where now each container receives at least three marbles but no more than nine. 3. Let an be the number of ways to select n marbles from a large supply of blue, red, and yellow marbles where the selection must include an even number of blue ones. Write down a generating function for the sequence an in closed form. 4. In this problem we are interested in counting the number of solutions to (+) a1 + a2 + az + 24 = n where a1, 22, 23, 24 > 0 In each part below express your generating function in a closed form. (a) Determine a generating function P1 (2) with the property that [x"]P1(x) is the number of solutions to (*). (b) Determine a generating function P2(2) with the property that "P2(2) is the number of solutions to (*) satisfying the additional condition that a, a, are even but az, 24 are odd. (c) Determine a generating function P3(2) with the property that "P3(x) is the number of solutions to *) satisfying the additional conditions that 10 < a < 100, az is even, and az is a multiple of 3. 5. Consider the set of six sided die where one dice has the numbers 1, 3, 4, 5, 6, 8 and the other has the numbers 1, 2, 2, 3, 3, 4 (see figure).

Answer 1

3) =) As we have given red, blue and yellow marbles Where selection must include an even number of blue ones. - For selection of even no. of blue marbles we can write it asa (ltre²trik txt-na). Now To select bote Bed and yellow there are no. conditions. we can write it as (l+r+ x² + x3t-.--) (ltra x² + x3t-on) - Forred For yellow. for. Therefore, generating function for problem can be written as f(3) = (1+2+x2+x+---){ (1+x++ x*+---) To write it in closed form, f(x)= a (x ) x + x F() = Consejo citas.

~ Note : - Formulas. Lotná z i+x+*+x+ - kx 1 p x + re + x + ---- na
That was very nice question :)