QUESTION 1
We want to evaluate
[tex] ^{10} C_3[/tex]
We use the formula,
[tex] ^{n} C_r = \frac{n!}{( n - r)!r!} [/tex]
Substitute
[tex]n = 10 \: and \: r = 3[/tex]
in to the above formula, to obtain,
[tex] ^{10} C_3= \frac{10!}{( 10 - 3)!3!} [/tex]
Simplify the right hand side to get,
[tex] ^{10} C_3= \frac{10!}{7!3!} [/tex]
This implies that,
[tex] ^{10} C_3= \frac{10 \times 9 \times 8 \times 7!}{7! \times 3 \times 2 \times 1} [/tex]
This simplifies to,
[tex] ^{10} C_3= \frac{10 \times 9 \times 8}{3 \times 2 \times 1} [/tex]
We cancel out common factors to get,
[tex] ^{10} C_3= \frac{10 \times 3 \times 4}{1 \times 1\times 1} [/tex]
[tex] ^{10} C_3= \frac{120}{1} [/tex]
[tex] \therefore \: ^{10} C_3=120[/tex]
QUESTION 2
We want to evaluate
[tex] ^{8} P_3[/tex]
We apply the formula,
[tex] ^{n} P_r =\frac{n!}{( n - r)!} [/tex]
We substitute
[tex]n = 8 \: and \: r = 3[/tex]
into the formula to get,.
[tex] ^{8} P_3=\frac{8!}{( 8 - 3)!} [/tex]
Simplify the right hand side to get,
[tex] ^{8} P_3=\frac{8!}{5!} [/tex]
This will further give us,
[tex] ^{8} P_3=\frac{8 \times 7 \times 6 \times 5!}{5!} [/tex]
This will Simplify to,
[tex] ^{8} P_3=\frac{8 \times 7 \times 6 \times 1}{1} [/tex]
[tex] ^{8} P_3=336[/tex]
