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Chapter 7: Problem 47

(a) Evaluate \(\left(\begin{array}{l}9 \\ 3\end{array}\right)\). (b) Evaluate \(\left(\begin{array}{l}9 \\ 6\end{array}\right)\).

Short Answer

Expert verified
(a) \(\left(\begin{array}{l}9 \\\ 3\end{array}\right) = 84\) (b) \(\left(\begin{array}{l}9 \\\ 6\end{array}\right) = 84\)

Step by step solution

01

Determine n and r for both combinations

For this exercise, we have: For (邪), n = 9 and r = 3 For (b), n = 9 and r = 6
02

Calculate factorials

Now, we will calculate the factorials for n, r and (n-r) for each combination: For (邪): n! = 9! = 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 362880 r! = 3! = 3 脳 2 脳 1 = 6 (n-r)! = (9-3)! = 6! = 720 For (b): n! = 9! = 362880 (We already calculated this for (邪)) r! = 6! = 720 (We already calculated this for (邪)) (n-r)! = (9-6)! = 3! = 6
03

Apply the combination formula

Using the combination formula, we will now evaluate both combinations: For (邪): \(\left(\begin{array}{l}9 \\\ 3\end{array}\right) = \frac{9!}{3!(9-3)!} = \frac{362880}{6 脳 720} = \frac{362880}{4320} = 84\) For (b): \(\left(\begin{array}{l}9 \\\ 6\end{array}\right) = \frac{9!}{6!(9-6)!} = \frac{362880}{720 脳 6} = \frac{362880}{4320} = 84\) So, the answers are: (a) \(\left(\begin{array}{l}9 \\\ 3\end{array}\right) = 84\) (b) \(\left(\begin{array}{l}9 \\\ 6\end{array}\right) = 84\)

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