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A First Course in Probability

Sheldon M. Ross

$Math Studyset Vaia Explanations$ Math

9th Edition · 432 pages

ISBN: 9780321794772

Chapter 1: Q. 1.24 (page 16)

Expand ${(3 x^{2} + y)}^{5}$ .

Short Answer

Expert verified

The value of ${(3 x^{2} + y)}^{5}$ is $y^{5} + 15 x^{2} y^{4} + 90 x^{4} y^{3} + 270 x^{6} y^{2} + 405 x^{8} y + 243 x^{10}$ .

Step by step solution

01

Step 1. Given information.

The given expression is ${(3 x^{2} + y)}^{5}$ .

02

Step 2. Expand the given expression.

Using the formula ${(a + b)}^{n} = \sum_{x = 0}^{n} (\begin{matrix} n \\ x \end{matrix}) {(a)}^{n - x} {(b)}^{x}$

${(3 x^{2} + y)}^{5} = \sum_{x = 0}^{5} (\begin{matrix} 5 \\ x \end{matrix}) {(3 x^{2})}^{5 - x} {(y)}^{x}$

$= (\begin{matrix} 5 \\ 0 \end{matrix}) {(3 x^{2})}^{5 - 0} {(y)}^{0} + (\begin{matrix} 5 \\ 1 \end{matrix}) {(3 x^{2})}^{5 - 1} {(y)}^{1} + (\begin{matrix} 5 \\ 2 \end{matrix}) {(3 x^{2})}^{5 - 2} {(y)}^{2} + (\begin{matrix} 5 \\ 3 \end{matrix}) {(3 x^{2})}^{5 - 3} {(y)}^{3} + (\begin{matrix} 5 \\ 4 \end{matrix}) {(3 x^{2})}^{5 - 4} {(y)}^{4} + (\begin{matrix} 5 \\ 5 \end{matrix}) {(3 x^{2})}^{5 - 5} {(y)}^{5} = 1 {(3 x^{2})}^{5} + 5 {(3 x^{2})}^{4} {(y)}^{1} + 10 {(3 x^{2})}^{3} {(y)}^{2} + 10 {(3 x^{2})}^{2} {(y)}^{3} + 5 {(3 x^{2})}^{1} {(y)}^{4} + 1 {(3 x^{2})}^{0} {(y)}^{5} = 243 x^{10} + 405 x^{8} y + 270 x^{6} y^{2} + 90 x^{4} y^{3} + 15 x^{2} y^{4} + y^{5}$

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Most popular questions from this chapter

Consider three classes, each consisting of $n$ students. From this group of $3 n$ students, a group of $3$ students is to be chosen.

(a) How many choices are possible?

(b) How many choices are there in which all $3$ students are in the same class?

(c) How many choices are there in which $2$ of the $3$ students are in the same class and the other student is in a different class?

(d) How many choices are there in which all $3$ students are in different classes?

(e) Using the results of parts (a) through (d), write a combinatorial identity.

Let $H_{k} (n)$ be the number of vectors $x_{1}, . . ., x_{k}$ for which each $x_{i}$ is a positive integer satisfying $1 \leq x_{i} \leq n$ and $x_{1} \leq x_{2} \leq, \dots, \leq x_{k} .$

(a)Without any computations, argue that

localid="1648218400232" $H_{1} (n) = n H_{k} (n) = \sum_{j = 1}^{n} H_{k - 1} (j) k > 1$

Hint: How many vectors are there in which $x_{k} = j$ ?

(b) Use the preceding recursion to compute $H_{3} (5)$ .

Hint: First compute $H_{2} (n) f o r n = 1, 2, 3, 4, 5$ .

Consider a tournament of $n$ contestants in which the outcome is an ordering of these contestants, with ties allowed. That is, the outcome partitions the players into groups, with the first group consisting of the players who tied for first place, the next group being those who tied for the next-best position, and so on. Let localid="1648231792067" $N (n)$ denote the number of different possible outcomes. For instance, localid="1648231796484" $N (2) = 3$ , since, in a tournament with localid="1648231802600" $2$ contestants, player localid="1648231807229" $1$ could be uniquely first, player localid="1648231812796" $2$ could be uniquely first, or they could tie for first.

(a) List all the possible outcomes when $n = 3$ .

(b) With localid="1648231819245" $N (0)$ defined to equal localid="1648231826690" $1$ , argue without any computations, that localid="1648281124813" $N (n) = \sum_{i = 1}^{n} (\begin{matrix} n \\ i \end{matrix}) N (n - i)$

Hint: How many outcomes are there in which localid="1648231837145" $i$ players tie for last place?

(c) Show that the formula of part (b) is equivalent to the following:

localid="1648285265701" $N (n) = \sum_{i = 1}^{n - 1} (\begin{matrix} n \\ i \end{matrix}) N (i)$

(d) Use the recursion to find N(3) and N(4).

For years, telephone area codes in the United States and Canada consisted of a sequence of three digits. The first digit was an integer between 2 and 9, the second digit was either 0 or 1, and the third digit was any integer from 1 to 9. How many area codes were possible? How many area codes starting with a 4 were possible

Seven different gifts are to be distributed among $10$ children. How many distinct results are possible if no child is to receive more than one gift?

See all solutions

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