Certainly! Let's break down the solution step by step for the given values [tex]\( n = 10 \)[/tex] and [tex]\( r = 4 \)[/tex].
### Step 1: Calculate Factorials
#### Calculate [tex]\( n! \)[/tex]
First, we need to find the factorial of [tex]\( n = 10 \)[/tex]:
[tex]\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800 \][/tex]
#### Calculate [tex]\( (n-r)! \)[/tex]
Next, we need to find the factorial of [tex]\( n-r \)[/tex]. Here, [tex]\( n-r = 10-4 = 6 \)[/tex], so we calculate [tex]\( 6! \)[/tex]:
[tex]\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \][/tex]
#### Calculate [tex]\( r! \)[/tex]
Finally, we find the factorial of [tex]\( r = 4 \)[/tex]:
[tex]\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \][/tex]
### Step 2: Evaluate [tex]\(\frac{n!}{(n-r)!}\)[/tex]
Using the calculated factorials:
[tex]\[ \frac{n!}{(n-r)!} = \frac{10!}{6!} \][/tex]
Plugging in the values we calculated:
[tex]\[ \frac{10!}{6!} = \frac{3,628,800}{720} = 5,040 \][/tex]
Hence,
[tex]\[ \frac{n!}{(n-r)!} = 5,040 \][/tex]
So, for [tex]\( n = 10 \)[/tex] and [tex]\( r = 4 \)[/tex],
[tex]\[ \frac{10!}{(10-4)!} = \frac{10!}{6!} = 5,040 \][/tex]
### Summary
(a) When [tex]\( n=10 \)[/tex] and [tex]\( r=4 \)[/tex],
[tex]\[ \frac{n!}{(n-r)!} = 5,040 \][/tex]
