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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Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.7: Distinguishable Permutations And Combinations

Problem 16E

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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

Expert Solution

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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