Annie has decided to stop eating Clif bars because they are

Annie has decided to stop eating Clif bars because they are so expensive. To wean herself, she will never eat more Clif bars on one day than she did the day before. In total, she plans to eat n Clif bars before stopping altogether. For example, if n = 10, here are a few possibilities (among others):
8,2

6,3,1

5,3,2

3,3,3,1

2,2,2,2,2,

3,1,1,1,1,1,1,1
Prove that the number of ways she can eat n Clif bars in at most k days, where k Z, is equal to the number of ways she can eat n Clif bars with at most k Clif bars per day.

Solution

There are 41 possibilties that Annie will never eat more Clif bars on one day than she did the day before are:

Now, Annie can eat 10 Clif bars in at the most 10 days if she were to eat one each day. So, k = 10.

Now if we were to find out the number of ways in which Annie can eat 10 Clif bars with at the most 10 bars per day, we will be arriving with the same 41 combinations for number of bars eaten on consecutive days. Because say she decides to go for the max say 10 bars in one day then she can accomplish her goal in exactly 1 day. If she goes for say

Annie can eat 10 Clif bars in at the most 10 days if she were to eat one each day. This she can do only in one way i.e. 1,1,1,1,1,1,1,1,1,1.

AThe number of days she can eat 10 Clif bars with at the most 10 Clif bars per day is also 1.