In Exercises 9–16, use the formula for nCr to evaluate each expre...

Question

In Exercises 9–16, use the formula for nCr to evaluate each expression.

9C5

Accepted Answer

Hello, today we're gonna be solving for the following combination. What we are given is the combination of seven and three. In order to solve for this combination, we need to use the formula for a general combination. So the general combination is given to us as the combination of N and R. And the formula for this combination is going to be N factorial over R factorial multiplied by the quant quantity N minus R factorial. So in order to use this formula, we need to identify our N and R values in our current combination, seven takes up the spot of N. So N is going to equal to seven in this case and three takes up the spot of R. So R is going to equal to three. And if we plug this in to the formula for a combination, we can rewrite the combination as N factorial which is seven factorial over R factorial, which is three factorial multiplied by the quantity of N minus R which is seven minus three factorial. So let's go ahead and simplify the denominator seven minus three is going to give us the value of four. So we have seven factorial over three factorial times four factorial. Now, in order to solve for a factorial, keep in mind that a factorial is the product of the current value multiplied by the decreasing numbers. So in this case here, if we wanted to expand three factorial, we can rewrite it as three times two times one. But there is a trick that we can do to simplify the denominator. Notice that in the numerator, we have a seven factorial. If we expand this term, we can expand it to give us seven times six times five times four factorial. And if we rewrite this numerator, using this statement, we can rewrite the combination as seven times six times five times four factorial over three factorial multiplied by four factorial. Now notice that we have a four factorial in both the numerator and denominator, we can go ahead and cancel them out for the numerator and denominator. And if we expand three factorial, the combination is going to expand to give us seven times six times 5/3 times two times one. Now, if we take the product of 76 and five in the numerator, we can rewrite the numerator as 210 and three times two times one of the denominator is going to simplify to give us six. Now 210 is divisible by 6, 210 divided by six is going to give us the value of 35. And this is going to be the value for the given combination. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 9–16, use the formula for nCr to evaluate each expression. 9C5

Key Concepts

Combination

Binomial Coefficient

Factorial

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