Discover a wealth of knowledge and get your questions answered on IDNLearn.com. Our platform provides detailed and accurate responses from experts, helping you navigate any topic with confidence.
Answer:
(a) [tex]1\, 663\, 200[/tex] ways to choose when the order matters.
(b) [tex]330[/tex] ways to choose when the order is not considered.
Step-by-step explanation:
The permutation formula gives the number of ways to choose and order [tex]k[/tex] items from a total of [tex]n[/tex] items ([tex]n \ge k[/tex]) without replacement:
[tex]\begin{aligned} {}_{n} P_{k} &= \frac{n!}{(n - k)!}\end{aligned}[/tex].
(The numerator is the factorial of [tex]n[/tex], while the denominator is the factorial of [tex](n - k)[/tex].)
The combination formula is for the case where the order within the [tex]k[/tex] selected items does not matter. The combination formula [tex]{}_{n} C_{k}[/tex] for choosing [tex]k[/tex] items out of a total of [tex]n[/tex] without replacement can be derived in the following steps:
In other words:
[tex]\begin{aligned} {}_{n} C_{k} &= \frac{{}_{n} P_{k}}{k!} && \genfrac{}{}{0em}{}{(\text{number of ordered choices})}{(\text{number of orderings within $k$ distinct items})} \\ &= \frac{n!}{(n - k)!\, (k)!}\end{aligned}[/tex].
In this question, [tex]n = 11[/tex] while [tex]k = 7[/tex].
The number of ways to choose when ordering matters (permutation) would be:
[tex]\begin{aligned} {}_{n} P_{k} &= \frac{n!}{(n - k)!} = \frac{11!}{(11 - 7)!} = 1\, 663\, 200\end{aligned}[/tex].
The number of ways to choose without considering ordering within the [tex]k[/tex] selected items (combination) would be:
[tex]\begin{aligned} {}_{n} C_{k} &= \frac{n!}{(n - k)!\, (k)!} = \frac{11!}{(11 - 7)!\, (7!)} = 330\end{aligned}[/tex].