Chapter 5.2, Problem 70E

(a)

To determine

Whether the provided expression is true or not.

Answer to Problem 70E

Solution: The provided expression is true.

Explanation of Solution

Calculation: Binomial coefficient is defined as the number of ways of ordering k successes among n observations. The formula is

$C{}_k=\frac{n!}{k!\left(n-k\right)!}\text{ , }k=0,1,2,\dots ,n$

Where,

$n!=n\times \left(n-1\right)\times \left(n-2\right)\times \cdots \times 3\times 2\times 1$

Now, by using binomial coefficient, the provided expression can be calculated as:

$\begin{array}{c}C{}_n=\frac{n!}{n!\left(n-n\right)!}\\ =\frac{n!}{n!\times 0!}\\ =\frac{n!}{n!\times 1}\\ =1\end{array}$

Hence, the provided expression is true.

To determine

To explain: The provided expression in words.

Answer to Problem 70E

Solution: From the above solution, it can be seen that there is only one way of distributing ‘n’ successes among ‘n’ trials.

Explanation of Solution

By using binomial coefficient, the provided expression shows the number of ways of ordering ‘n’ successes among ‘n’ observations. Hence, from the above solution, it can be seen that there is only one way of ordering ‘n’ successes among ‘n’ observations.

(b)

To determine

To test: Whether the provided expression is true or not.

Answer to Problem 70E

Solution: The provided expression is true.

Explanation of Solution

Calculation: By using binomial coefficient, the provided expression can be calculated as:

$\begin{array}{c}C{}_{n-1}=\frac{n!}{\left(n-1\right)!\left\{n-\left(n-1\right)\right\}!}\\ =\frac{n!}{\left(n-1\right)!\times 1!}\\ =\frac{n\times \left(n-1\right)!}{\left(n-1\right)!\times 1}\\ =n\end{array}$

Conclusion: Hence, the provided expression is true.

To determine

To explain: The provided expression in words.

Answer to Problem 70E

Solution: From the above solution, it can be seen that there are only n ways of distributing $″n-1″$ successes among ‘n’ trials.

Explanation of Solution

By using binomial coefficient, the provided expression is shows the number of ways of ordering $″n-1″$ successes among ‘n’ observations. Hence, from the above solution, it can be seen that there are n ways of ordering $″n-1″$ successes among n observations.

(c)

To determine

To test: Whether the provided expression is true or not.

Answer to Problem 70E

Solution: The provided expression is true.

Explanation of Solution

Calculation: By using binomial coefficient, the provided expression can be calculated as:

L.H.S,

$C{}_k=\frac{n!}{k!\left(n-k\right)!}$

R.H.S,

$\begin{array}{c}C{}_{n-k}=\frac{n!}{\left(n-k\right)!\left\{n-\left(n-k\right)\right\}!}\\ =\frac{n!}{\left(n-k\right)!\left\{n-n+k\right\}!}\\ =\frac{n!}{\left(n-k\right)!k!}\end{array}$

From the above solution, it can be seen that L.H.S = R.H.S. Hence, the provided expression is true.

To determine

To explain: The provided expression in words.

Answer to Problem 70E

Solution: From the above solution, it can be seen that the number of ways of distributing ‘k’ successes to ‘n’ trials is exactly equal to the number of ways of distributing $″n-k″$ successes to ‘n’ trials.

Explanation of Solution

By using binomial coefficient, the provided expression $C{}_k$ is shows the number of ways of ordering ‘k’ successes among ‘n’ observations and the expression $C{}_{n-k}$ is defined as the number of ways of ordering $″n-k″$ successes among ‘n’ observations. Hence, from the above solution, it can be seen that the number of ways of ordering ‘k’ successes among ‘n’ observations is equal to the number of ways of ordering $″n-k″$ observations among ‘n’ observations, that is, $C{}_k=C{}_{n-k}$.