Let's find the values of [tex]\( \frac{n!}{(n-r)!} \)[/tex] and [tex]\( \frac{n!}{r!(n-r)!} \)[/tex] given [tex]\( n = 10 \)[/tex] and [tex]\( r = 4 \)[/tex].
### Part (a)
Evaluate [tex]\( \frac{n!}{(n-r)!} \)[/tex]:
1. Substitute [tex]\( n = 10 \)[/tex] and [tex]\( r = 4 \)[/tex] into the expression:
[tex]\[ \frac{10!}{(10-4)!} = \frac{10!}{6!} \][/tex]
2. Write out the factorials:
[tex]\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
[tex]\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
3. Cancel out the common factorial terms (from 6!):
[tex]\[ \frac{10 \times 9 \times 8 \times 7 \times 6!}{6!} = 10 \times 9 \times 8 \times 7 \][/tex]
4. Calculate the product:
[tex]\[ 10 \times 9 = 90 \][/tex]
[tex]\[ 90 \times 8 = 720 \][/tex]
[tex]\[ 720 \times 7 = 5040 \][/tex]
Therefore, [tex]\( \frac{10!}{6!} = 5040 \)[/tex].
### Part (b)
Evaluate [tex]\( \frac{n!}{r!(n-r)!} \)[/tex]:
1. Substitute [tex]\( n = 10 \)[/tex] and [tex]\( r = 4 \)[/tex] into the expression:
[tex]\[ \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!} \][/tex]
2. Write out the factorials:
[tex]\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \][/tex]
[tex]\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \][/tex]
[tex]\[ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \][/tex]
3. Substitute these values into the expression:
[tex]\[ \frac{10!}{4! \cdot 6!} = \frac{10!}{24 \cdot 720} \][/tex]
4. Note that [tex]\( 10! = 5040 \times 720 \)[/tex]. This allows you to cancel 720:
[tex]\[ \frac{5040 \times 720}{24 \times 720} = \frac{5040}{24} \][/tex]
5. Calculate the division:
[tex]\[ 5040 \div 24 = 210 \][/tex]
Therefore, [tex]\( \frac{10!}{4! \cdot 6!} = 210 \)[/tex].
### Summary
(a) When [tex]\( n = 10 \)[/tex] and [tex]\( r = 4 \)[/tex], [tex]\( \frac{n!}{(n-r)!} = 5040 \)[/tex].
(b) When [tex]\( n = 10 \)[/tex] and [tex]\( r = 4 \)[/tex], [tex]\( \frac{n!}{r!(n-r)!} = 210 \)[/tex].
