1 00:00:00,757 –> 00:00:02,118 hello and welcome.

2 00:00:02,186 –> 00:00:05,649 The mathematical symbol of today is the binomial coefficient.

3 00:00:05,849 –> 00:00:07,336 Written with parentheses.

4 00:00:08,529 –> 00:00:13,388 and there at the top we find an integer n and at the bottom an integer k.

5 00:00:14,100 –> 00:00:18,647 Now usually you read this binomial coefficient as “n choose k”.

6 00:00:19,257 –> 00:00:27,480 We do this, because it represents the number of possibilities to choose k elements from a fixed set with n elements.

7 00:00:28,529 –> 00:00:32,721 Here i always think it’s a good idea to visualize this with an example.

8 00:00:33,443 –> 00:00:37,783 So here you see we have 7 balls. So n is equal to 7.

9 00:00:38,771 –> 00:00:43,943 and now i tell you we take 3 of them out. So k is equal to 3.

10 00:00:44,971 –> 00:00:49,640 Here maybe one possibility would be to get 2, 3 and 6.

11 00:00:50,314 –> 00:00:52,756 and please note here there is no order involved.

12 00:00:52,786 –> 00:00:55,135 So we take them out with one pick.

13 00:00:55,943 –> 00:00:59,000 Of course still there are a lot of other examples.

14 00:00:59,157 –> 00:01:01,216 Maybe 5, 6 and 7.

15 00:01:01,971 –> 00:01:07,586 and now the question the binomial coefficient answers, is how many possibilities there are.

16 00:01:08,629 –> 00:01:12,097 Therefore i would suggest trying to calculate them.

17 00:01:12,729 –> 00:01:16,629 So here you see 3 empty slots that we can fill in with numbers.

18 00:01:17,929 –> 00:01:23,431 Now if we just take out one ball then we could have any number here in the first position.

19 00:01:24,643 –> 00:01:28,655 Hence counting the possibilities we have exactly n here.

20 00:01:29,429 –> 00:01:32,414 Then for the second position we take out another ball.

21 00:01:33,629 –> 00:01:39,400 However now the first ball is already gone. So we have exactly (n - 1) numbers.

22 00:01:40,414 –> 00:01:44,486 and finally for the third position (n - 2) numbers remain.

23 00:01:45,671 –> 00:01:53,639 Now multiplying these numbers gives us exactly the possibilities for filling in these slots with the balls here.

24 00:01:54,886 –> 00:01:58,485 However then you might see we have an order involved.

25 00:01:59,629 –> 00:02:05,542 Therefore we have to divide this thing here by the number of orders we have for these slots.

26 00:02:06,429 –> 00:02:09,670 Indeed the reasoning now is similar to before.

27 00:02:10,471 –> 00:02:15,655 If you want to rearrange the first ball, the first slot, you have 3 positions to choose.

28 00:02:16,586 –> 00:02:21,137 Then for the second only 2 remain and for the last one only 1 remains.

29 00:02:22,486 –> 00:02:26,400 and with this we have it. This is what the binomial coefficient should do.

30 00:02:27,243 –> 00:02:37,015 So in general here we would have: n times (n - 1) times (n - 2) and so on until we reach (n - k + 1).

31 00:02:37,971 –> 00:02:41,709 So you should see this is the generalisation of our example.

32 00:02:42,529 –> 00:02:46,510 and moreover then we would find k! in the denominator.

33 00:02:47,657 –> 00:02:53,213 So indeed this is the definition of the binomial coefficient “n choose k”.

34 00:02:53,943 –> 00:02:57,671 and for the end i can tell you there is a shorter definition for this.

35 00:02:58,586 –> 00:03:07,439 Namely we have n! divided by k! as before and also (n - k)!.

36 00:03:08,386 –> 00:03:13,025 There you should see these 2 parts here just cancel to get you this.

37 00:03:13,771 –> 00:03:16,057 Ok so this is the binomial coefficient.

38 00:03:16,400 –> 00:03:18,500 and i hope that you learned something today.

39 00:03:19,386 –> 00:03:21,002 Then see you next time.

40 00:03:21,202 –> 00:03:21,829 Bye!

Q1: What is $\binom{n}{1}$?

A1: $n$

A2: $n-1$

A3: $n!$

A4: $\frac{n!}{2!}$

A5: $n(n-1)$

Q2: What is $\binom{n}{n}$?

A1: $1$

A2: $n-1$

A3: $n!$

A4: $\frac{n!}{2!}$

A5: $0$

Q3: Let $n,k \in \mathbb{N}$ with $n \leq k$. Is $\binom{n}{k} = \binom{n}{n-k}$ correct?

A1: Yes, always!

A2: No, never!

A3: Only for $n = k = 1$