Stars and Bars: all problems

How many ways are there to distribute 5 identical candies to 3 children?

When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as

where $n$ is a positive integer. What is $n$?

Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?

Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?

Three squirrels have to distribute 19 identical acorns amongst themselves.

• The first squirrel demands to have a positive odd number of acorns.
• The second squirrel is fine with any non-negative even number of acorns.
• The third squirrel insists he must get an even number of acorns greater than or equal to 4.

How many different ways can they distribute the acorns?

How many positive 5 digit integers exist such that each digit is greater than or equal to the previous digit?

Find the number of solutions to $a + b + c + d < 19$ where $a, b, c, d$ are positive integers, $a$ and $c$ must be odd, and $b$ and $d$ must be even.