Binomial coefficients in practice

Hello. Let us see the first exercise of the binomial coefficient in practice step, always into week 3. We want to count in how many ways one can choose 3 cards in a deck of 10 cards. Ok, when you have to choose without any particular order, elements in a set of different elements, you have to use the binomial coefficients. In particular, for this exercise, we have that the number of ways in which we can choose 3 cards in a deck of 10 cards is exactly equal to the binomial 10 over 3. This number– now we’ll see what is this number. But this symbol is exactly the number of possible choices of 3 elements among 10 elements.

This is by definition equal to the factorial of 10 over the factorial of 10 minus 3 times the factorial of 3. Now, we can simplify the factorial of 10 and the factorial of 7, and what remains is exactly 10 times 9 times 8. And at the denominator, we have only the factorial of 3, which is 3 times 2, if you want, times 1, but this, of course, is not necessary. And now what do we get? You easily can simplify this fraction, 3 and 9, and you get 3 here and 2 and 8, and you get 4 here. And what you get? 10 times 3, which are 30 times 4, which is 120.

Then we can choose 3 cards in a deck of 10 cards in 120 different ways.

Let us see now the second exercise of the binomial coefficients in practice step. We have to prove that the binomial coefficient, n over k, is equal to the binomial coefficient n over n minus k for any k and n, which satisfy these equalities. Then let us start to with first just considering the definition of the binomial coefficient. You know that n over k is equal by definition to the factorial of n over the factorial of n minus k times the factorial of k. And what is the definition of this other binomial coefficient? You have that n over n minus k is equal to the factorial of n.

And in the denominator, we have the factorial of the difference between n and n minus k– n minus n minus k factorial– times the factorial of n minus k.

But this is exactly n factorial. And here what you have, n minus n minus k, and you remain with k factorial times n minus k factorial. And this and this are perfectly equal. But this is just an algebraic proof, if you want. But it’s much more interesting to consider a proof which regards the meaning of the binomial coefficient in a combinatorial sense. Remember, what means n over k? This number counts in how many ways you can choose k elements among n different elements. That is, you have n different elements, and you want to choose k elements among these n. For example, you can decide to choose these k elements among these n elements.

In general, in how many ways can you choose k elements among n, binomial coefficient n over k times. But observe. What means to choose these k elements among these n elements is exactly like to decide to discard these n minus k elements. Just no. You put away these n minus k element, and you remain with your k elements. Therefore, to choose k elements among n is equivalent to discard n minus k elements among your n elements. Then to choose k elements among n, and you can do the same exactly binomial coefficient n over k times– is equal to choose the n minus k element to discard among your n elements. And therefore, these two binomial coefficients are equal. Ciao.