# Understanding Binomial Coefficients: n Choose r

• 2^Oscar
In summary, the conversation discusses the concept of binomial coefficients and n choose r, with a focus on understanding the proof for (n choose r) = n!/(r!(n-r)!). The proof involves three steps, including choosing an r-element subset of a set S, arranging the chosen elements, and arranging the remaining elements of S. The conversation also touches on the idea of identical elements and how they affect the total number of arrangements. The conversation concludes with the understanding that this concept is related to multinomial coefficients.
2^Oscar
Hey guys,

I've been reading up on binomial coefficients and I have found a brief section on n choose r. I understand vaguely what it actually is, however in my textbook there is a step by step proof of how we show that:

( $$\stackrel{n}{r}$$ ) = $$\frac{n!}{r!(n-r)!}$$

I can follow where this comes from. My book states that S={1,2...n} and then proceeds to prove the above in three steps; firstly by choosing an r-element subset (called T) of S where there are ( $$\stackrel{n}{r}$$ ) choices. Secondly choosing an arrangement of T where there will be r! choices. Finally by choosing an arrangement of the remaining n-r elements of S where there are (n-r)! choices. By the multiplication principle the total number of arrangements of S (i.e. n!) is equal to the product of all these three. Then you simply rearrange your result to get the above.

Could someone please explain to me the second and third steps of this because I'm really struggling to see how these combined with step one would give the number of arrangements of S.

Cheers,
Oscar

Well, here's a different way of saying the same thing. "n choose r" means that you have a set of n objects and you are choosing r of those objects. How many ways can you do that? One method is this: imagine that you have all n objects in a row before you. Write, below each one "Y" if you are choosing it, "N" if not. Now you have n letters, r of them "Y" and n-r of them "N".

How many different ways are there of ordering (or writing) a word consisting of r "Y"s and n-r "N"s? Okay, now imagine labeling each of the "Y"s "1", "2", ..., "r" and each of the n-r "N"s "1", "2", ..., "n-r" so that they are all different. How many ways can you order n different symbols? Of course, you have n choices for the first, then, having chosen that, n-1 symbols are left so you have n-1 choices for the second, then n-2 for the third, etc. There are n(n-1)(n-1)...(3)(2)(1)= n! ways to do that.

But, of course, the "Y"s are NOT all different. For anyone of those n! arrangements, we could swap two "Y"s and not change it. How many different ways are there to swap just the "Y"s? Since there are r "Y"s there are r! ways to do that. That is, every arrangement with the identical "Y"s has been counted r times. To find the actual number of ways of arranging those letters, we have to divide by r! to allow for the rearrangements of just the "Y"s. Similarly, because there are n-r identical "N"s, we have to divide by (n-r)! to allow for that: altogether, the number of ways to arrange n objects, n of them identical and the other n-r also identical is
$$\frac{n!}{r!(n-r)!}$$

I see exactly what you mean now, part of what helped was reading the next chapter about multinomial coefficients; it helps put things in perspective.

Thanks again,
Oscar

## 1. What are binomial coefficients?

Binomial coefficients, also known as the "choose" function, are mathematical values that represent the number of ways to choose a specific number of objects from a larger set of objects. They are denoted by the symbol "n choose r" or nCr, where n represents the total number of objects and r represents the number of objects to be chosen.

## 2. How do you calculate binomial coefficients?

The formula for calculating binomial coefficients is nCr = n! / (r!(n-r)!), where n! represents n factorial (the product of all positive integers from 1 to n). Alternatively, you can use a combination calculator or Pascal's triangle to find the values of binomial coefficients.

## 3. What is Pascal's triangle and how is it related to binomial coefficients?

Pascal's triangle is a triangular arrangement of numbers that starts with a 1 at the top, and each subsequent row is formed by adding the two numbers directly above it. The values in each row of Pascal's triangle correspond to the coefficients in the binomial expansion of (a + b)^n, where n is the row number. This relationship is known as the binomial theorem.

## 4. How are binomial coefficients used in probability and statistics?

Binomial coefficients are used to calculate probabilities in situations where there are only two outcomes (success or failure) and a fixed number of trials. This is known as a binomial experiment. In statistics, binomial coefficients are used in the binomial distribution, which is a probability distribution that describes the likelihood of a certain number of successes in a given number of trials.

## 5. Can binomial coefficients be negative?

No, binomial coefficients cannot be negative. This is because they represent the number of ways to choose objects, and the number of ways cannot be negative. If the formula for calculating binomial coefficients results in a negative value, it is typically interpreted as zero, as there are no ways to choose a negative number of objects.

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