Sure! Let's evaluate the expressions step by step given the values [tex]\( n = 13 \)[/tex] and [tex]\( r = 4 \)[/tex].
### Part (a)
We need to evaluate [tex]\(\frac{n!}{(n-r)!}\)[/tex].
1. Substitute the values of [tex]\( n \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[
n = 13, \quad r = 4
\][/tex]
2. Calculate [tex]\( n - r \)[/tex]:
[tex]\[
n - r = 13 - 4 = 9
\][/tex]
3. The expression simplifies to:
[tex]\[
\frac{13!}{9!}
\][/tex]
4. Remember that [tex]\( 13! = 13 \times 12 \times 11 \times 10 \times 9! \)[/tex], so [tex]\(\frac{13!}{9!}\)[/tex] simplifies to:
[tex]\[
\frac{13 \times 12 \times 11 \times 10 \times 9!}{9!} = 13 \times 12 \times 11 \times 10
\][/tex]
5. Perform the multiplications:
[tex]\[
13 \times 12 = 156
\][/tex]
[tex]\[
156 \times 11 = 1716
\][/tex]
[tex]\[
1716 \times 10 = 17160
\][/tex]
So, the value of [tex]\(\frac{n!}{(n-r)!}\)[/tex] when [tex]\( n = 13 \)[/tex] and [tex]\( r = 4 \)[/tex] is:
[tex]\[
\boxed{17160}
\][/tex]
### Part (b)
We need to evaluate [tex]\(\frac{n!}{r!(n-r)!}\)[/tex].
1. Again, substitute the values of [tex]\( n \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[
n = 13, \quad r = 4
\][/tex]
2. Calculate [tex]\( n - r \)[/tex]:
[tex]\[
n - r = 13 - 4 = 9
\][/tex]
3. The expression simplifies to:
[tex]\[
\frac{13!}{4! \cdot 9!}
\][/tex]
4. We already know from Part (a) that:
[tex]\[
\frac{13!}{9!} = 13 \times 12 \times 11 \times 10 = 17160
\][/tex]
5. Now, find the factorial of [tex]\( r = 4 \)[/tex]:
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
6. Divide the result from Part (a) by [tex]\( r! \)[/tex]:
[tex]\[
\frac{17160}{24}
\][/tex]
7. Perform the division:
[tex]\[
17160 \div 24 = 715
\][/tex]
So, the value of [tex]\(\frac{n!}{r!(n-r)!}\)[/tex] when [tex]\( n = 13 \)[/tex] and [tex]\( r = 4 \)[/tex] is:
[tex]\[
\boxed{715}
\][/tex]
