↪20 points
Expand using binomial theorem:
Answers
(
a
+
b
)
6
=
a
6
+
6
a
5
b
+
15
a
4
b
2
+
20
a
3
b
3
+
15
a
2
b
4
+
6
a
b
5
+
b
6
Why does this work?
If we write out the value as a product of binomials we have:
(
a
+
b
)
6
=
(
a
+
b
)
(
a
+
b
)
(
a
+
b
)
(
a
+
b
)
(
a
+
b
)
(
a
+
b
)
If you pick one term from each binomial and multiply them together, then you have made
6
choices of left or right. If you choose all left, then you end up with
a
6
- which you can do just one way. If you choose all right then you end up with
b
6
, which again you can only do one way.
Otherwise, you are making a mixture of left and right choices, analogous to picking your way down from the top of Pascal's triangle to the bottom, via a sequence of left and right branches. The power of
a
resulting is the number of left branches you choose and the power of
b
the number of right branches.
Each number in Pascal's triangle is the sum of the two above, each of which counts the number of ways to reach that point by a sequence of left and right choices. So all of the numbers in Pascal's triangle count the number of ways to reach them by left/right choices starting at the top.
Binomial theorem :
For any positive integer n,
Solution :
Now,
[ You can also check the given Pascal's triangle from the given attachment for clarification. ]
