Due Friday, July 24
1. Prove the hockeystick identity r
X n + k k k =0
= n + r + 1 r when n, r ≥ 0 by
(a) using a combinatorial argument. (You want to choose r objects. For each k: choose the first r − k in a row, skip one, then how many choices do you have for the remaining objects?)
For each k , choose the first r − k objects in a row, then skip one (so you choose exactly r − k n + r + 1 − ( r − k + 1) from the front of the line), then you have = n + k options for the r − ( r − k ) k remaining objects. Summing from k = 0 to r gives P r k =0 n + k k
= n + r + 1 r
, the total number of ways to choose r objects from n + r + 1.
(b) Using Pascal’s identity (plus induction! For the induction, fix n arbitrarily and then use induction on r ) m n n + 1
Fix n arbitrarily. For r = 0 this is just = , which is true since
0 0
For the inductive step: suppose that the formula holds for r , then
0
= 1 for any m .
r +1
X k =0 n + k k
= r
X n + k k k =0 n + r + 1
= r
= n + r + 2 r + 1
+ n + r + 1 r + 1
+ n + r + 1 r + 1
So if the formula holds for r then it also holds for r + 1. Since n was picked arbitrarily, the formula works for any n and r ≥ 0.
2. How many ways can you split nine people into three groups of three?
There are
9!
3!3!3!
= 1680 ways to split people into three distinct groups, then 6 more ways to shuffle the groups around, so 280 ways in total.
Alternately, number the friends from 1 to 9. There are
8
2 ways to pick two people to be on person
1’s team, then
5
2 ways to pick two people to be on the team with the person with the smallest remaining number. The final three are on a team, giving
8
2
·
5
2
= 280 options.
3. How many ways can you give nine (identical) cookies to three friends? It is okay to give no cookies to some of those friends.
9 + 2 11
Stars and bars problem: the total number of ways is = = 55.
2 2
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4. Remember that problem with the spheres and the triangular pyramid from an earlier homework? Here it is again!
(a) Use the hockeystick identity to show that T n
= 1 + 2 + . . .
+ n is equal to n + 1
2
.
Follows immediately from n = n
1
.
(b) Then use the hockeystick identity to show that T
1
+ T
2
+ . . .
+ T n
= n + 2
3
.
Follows immediately from T n
= n + 1
2
.
(c) Give an illustration of the above using Pascal’s triangle.
It looks like a line going downward parallel to the right side of the triangle, then with a small leftward hook for the last element.
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