Evaluate the given expression. 11 C 7 _{11}C_7

In this problem, we're asked to evaluate the given expression. And the expression we're given is 11 c 17 or the number of combinations of 11 things taken 7 at a time. So since we're already given it in this combinations notation, all that's left to do is plug this into the combinations formula. So let's go ahead and do that. Now we have 11c7 and we can set that equal to our formula plugging in our values for n and r. Now our value for n is that first number 11 and then our value for r is that second number 7. So I start out with 11 factorial on the top and then on the bottom, I take n minus r, so in this case, 11 minus 7 factorial, and then multiply that by r factorial, in this case 7 factorial. Now we can go ahead and start simplifying this, leaving our numerator as 11 factorial for now and then taking our denominator, 11 minus 7 is going to give me 4. So this is 4 factorial times 7 factorial. Now we want to go ahead and rewrite that numerator in order to cancel the highest factorial in the denominator, which in this case is 7 factorial. So let's go ahead and rewrite that numerator, expanding that 11 factorial out. So this is 11 times 10 times 9 times 8 times 7 factorial, which is what we want to cancel, and then this all gets divided by 4 factorial times 7 factorial. Now my 7 factorial will cancel on the top and the bottom. And then in my numerator, I'm left with 11 times 10 times 9 times 8, and then in my denominator I'm left with 4 factorial which I can expand out to 4 times 3 times 2 times 1. Now multiplying this out for my numerator I end up with 7,920 and then in my denominator I'm left with 4 factorial which is just 24. Now fully dividing that out gives me my final answer of 330 that represents my given expression. Thanks for watching and let me know if you have any questions.

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Combinatorics

College Algebra

10. Combinatorics & Probability

Combinatorics

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Multiple Choice

Evaluate the given expression. $sub 11 C sub 7$

330

120

5040

7920

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Verified step by step guidance

Understand that the expression 11C7 represents a combination, which is calculated using the formula $ \binom{n}{r} = \frac{n!}{r!(n-r)!} $. Here, $ n = 11 $ and $ r = 7 $.

Calculate the factorial of 11, denoted as $ 11! $, which is the product of all positive integers up to 11.

Calculate the factorial of 7, denoted as $ 7! $, which is the product of all positive integers up to 7.

Calculate the factorial of $ 11 - 7 = 4 $, denoted as $ 4! $, which is the product of all positive integers up to 4.

Substitute these factorial values into the combination formula $ \binom{11}{7} = \frac{11!}{7! \cdot 4!} $ and simplify the expression to find the number of combinations.