In Exercises 7-14, perform the indicated calculation.
12...

Question

In Exercises 7-14, perform the indicated calculation.

12. (10C7)/(14C7)

Accepted Answer

Welcome back, everyone. Evaluate the following combination. 8, choose 3, divided by 12, choose 3. A says 55 divided by 14, B, 14 divided by 55, C 7 divided by 110, and D 56 divided by 55. For this problem, let's recall the formula for combinations and choose K. So we take N factorial. Divided by K factorial, which is multiplied by N minus K factorial. And we're going to apply this formula for our combinations. So we have 8, choose 3 divided by 12, choose 3. Now, let's apply the formula for our numerator. Our N is 8, so we take 8 factorial, and we divide by K factorial, that will be 3 factorial, which is multiplied by 8 minus 3 factorial. Then in our denominator, we have 12 for our n, so there'd be 12 factorial divided by k factorial or simply speaking 3 factorial, which is multiplied by 12 minus 3 factorial. And that we can simplify. So first of all, we have a factorial. Divided by 3 factorial, which is multiplied by 5 factorial. And because we're dividing by a fraction, we can just multiply by the reciprocal of the bottom fraction, right? So we're multiplying by 3 factorial. Which is multiplied by 9 factorial, and dividing by 12 factorial. We can see that we can cross out 3 factorial because it belongs to the opposite sides of the resultant fraction, and now we get 8 factorial multiplied by 9 factorial. Divided by 5 factorial, multiplied by 12 factorial. And now let's go ahead and simplify. So 8 factorial is simply 8 multiplied by 76, and 5 factorial. We're dividing that by 5 factorial. That would be our denominator. And then multiplying by 9 factorial divided by 12 factorial, right? So, that's 9 factorial divided by. 12 factorial can be expressed as 12 multiplied by 1110, and 9 factorial. Now, this way we can cross out 5 factorial and 9 factorial. And we can see that we get 8 multiplied by 7 multiplied by 6. Divided by 12, multiplied by 11, multiplied by 10. So now let's perform the calculations. And to simplify them as much as possible, we can also cross out 6. And instead of 12 in the denominator, we're going to get 2, right? Because our numerator and our denominator, they're both divisible by 6. Now, considering 8 and 10, we can divide those by 2 as well. So 8 becomes 4 and 10 becomes 5. So now we get 4 multiplied by 7, which is 28, that's our enumerator. And our denominator, that will be 2 multiplied by 11 multiplied by 5. So, overall, we get 110. And now let's divide both sides by 2, so that'll be 14 divided by 55, which corresponds to the answer choice B. Thank you for watching.

In Exercises 7-14, perform the indicated calculation.
12. (10C7)/(14C7)

Key Concepts

Combinations

Factorial

Ratio of Combinations

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