Use generating functions to find the number of ways to select 10 balls from alarge pile of red, white, and blue balls if(a) The selection has at least two balls of each color(b) The selection has at most two red balls(c) The selection has an even number of blue balls

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Description: Use generating functions to find the number of ways to select 10 balls from alarge pile of red, white, and blue balls if(a) The selection has at least two balls of each color(b) The selection has at most two red balls(c) The selection has an even nu

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Use generating functions to find the number of ways to select 10 balls from a
large pile of red, white, and blue balls if
(a) The selection has at least two balls of each color
(b) The selection has at most two red balls
(c) The selection has an even number of blue balls

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Step 1

(a) Now since here we do not want less than two balls the coefficient for x⁰ and x¹ for the generating function for the red balls is zero and 1 for the others.

The generating function for at least two red balls.

$0 x^{0} + 0 x^{1} + x^{2} + x^{3} + x^{4} + . . . = x^{2} + x^{3} + x^{4} + . . .$

Similarly the generating function for at least two white balls is;

$x^{2} + x^{3} + x^{4} + . . .$

and for at least two blue balls is;

$x^{2} + x^{3} + x^{4} + . . .$

Hence the generating function for at least two balls for each color;

$(x^{2} + x^{3} + x^{4} + . . .) \times (x^{2} + x^{3} + x^{4} + . . .) \times (x^{2} + x^{3} + x^{4} + . . .) = {(x^{2} + x^{3} + x^{4} + . . .)}^{3} = {[x^{2} (1 + x^{2} + x^{2} + . . .)]}^{3} = x^{6} {(1 + x^{2} + x^{2} + . . .)}^{3}$

Hence the number of ways is the coefficient of x¹⁰ , which is;

${}^{(10 - 6) + 3 - 1}C_{10 - 6} = {}^{6}C_{4} = 15 w a y s$

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Use generating functions to find the number of ways to select 10 balls from a large pile of red, white, and blue balls if (a) The selection has at least two balls of each color (b) The selection has at most two red balls (c) The selection has an even number of blue balls