Find the following combinations ,C, (a) n= 8 andr=2. 8C2 (b) n=8 and r D1. BC1 (c) n=8 and r 6. 8C6 (d) n= 8 andr38. 8C8

MATLAB: An Introduction with Applications
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Author:Amos Gilat
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**Find the following combinations \( nC_r \):**

**(a) \( n = 8 \) and \( r = 2 \)**
- \( 8C2 \) [Input box available to enter the answer]

**(b) \( n = 8 \) and \( r = 1 \)**
- \( 8C1 \) [Input box available to enter the answer]

**(c) \( n = 8 \) and \( r = 6 \)**
- \( 8C6 \) [Input box available to enter the answer]

**(d) \( n = 8 \) and \( r = 8 \)**
- \( 8C8 \) [Input box available to enter the answer]

---

### Explanation:

This section requires calculating combinations, denoted as \( nC_r \), which are used to determine the number of ways to choose \( r \) items from a total of \( n \) items without regard to the order of selection. The formula to calculate combinations is:

\[
nC_r = \frac{n!}{r!(n-r)!}
\]

Where \( n! \) (n factorial) is the product of all positive integers up to \( n \).

To solve each of the problems:

- **(a)** \( 8C2 \): Calculate how many ways to choose 2 items from 8.
- **(b)** \( 8C1 \): Calculate how many ways to choose 1 item from 8.
- **(c)** \( 8C6 \): Calculate how many ways to choose 6 items from 8.
- **(d)** \( 8C8 \): Calculate how many ways to choose all 8 items from 8. 

By using the provided formula, the required values of combinations can be determined for each case.
Transcribed Image Text:**Find the following combinations \( nC_r \):** **(a) \( n = 8 \) and \( r = 2 \)** - \( 8C2 \) [Input box available to enter the answer] **(b) \( n = 8 \) and \( r = 1 \)** - \( 8C1 \) [Input box available to enter the answer] **(c) \( n = 8 \) and \( r = 6 \)** - \( 8C6 \) [Input box available to enter the answer] **(d) \( n = 8 \) and \( r = 8 \)** - \( 8C8 \) [Input box available to enter the answer] --- ### Explanation: This section requires calculating combinations, denoted as \( nC_r \), which are used to determine the number of ways to choose \( r \) items from a total of \( n \) items without regard to the order of selection. The formula to calculate combinations is: \[ nC_r = \frac{n!}{r!(n-r)!} \] Where \( n! \) (n factorial) is the product of all positive integers up to \( n \). To solve each of the problems: - **(a)** \( 8C2 \): Calculate how many ways to choose 2 items from 8. - **(b)** \( 8C1 \): Calculate how many ways to choose 1 item from 8. - **(c)** \( 8C6 \): Calculate how many ways to choose 6 items from 8. - **(d)** \( 8C8 \): Calculate how many ways to choose all 8 items from 8. By using the provided formula, the required values of combinations can be determined for each case.
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