In Exercises 1–8, evaluate the given binomial coefficient.

Question

Binomial coefficient notation for exercise 1: (8 choose 3)

Accepted Answer

hey everyone here we are asked to find the value of the binomial coefficient for the expression 12 7. Now before we solve this expression here we need to recall the formula for calculating a binomial coefficient which has the expression and K equal to n factorial divided by K factorial times the quantity of n minus K factorial. So now we just need to look at our given expression and identify the N and K values. So our top value is 12. So we know that N is going to be 12 and then the lower value is seven. Therefore our K is equal to seven. And now with these values clearly identified, we can go ahead and just plug them into our formula and solve for our coefficient. So we see that the expression 12 7 is equal to in the numerator we have n factorial which is factorial. And then we have seven factorial for our K factorial times the quantity of N which is minus K, which is seven Factorial. And now we can just start simplifying. So keeping the numerator the same. For now we have 12 factorial divided by seven factorial times 12 -7 gives us five factorial. And now with this seven factorial in our denominator the larger factorial of the two. We can expand our numerator factorial until we also have a seven factorial. So doing this in the numerator we have 12 times times 10 times nine times eight times seven factorial. All divided by seven factorial times five factorial. And now we can just cross out or cancel out our seven factorial since they're both in the numerator and the denominator. And then we are left with 12 times 11 times 10 times nine times eight in our numerator and in the denominator we have five factorial which expands to five times four times three times two times one. And now multiplying these terms together and simplifying, we see that our binomial coefficient is 792. Leaving us with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

In Exercises 1–8, evaluate the given binomial coefficient.

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