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Distribution into Bins

How can five candies be distributed among 3 friends? Well, the answer depends on if you can tell the candies apart!

Level 2

         

There are 5 stars and 4 bars to be arranged in the 9 spaces pictured below. One possible arrangement is shown.

How many possible arrangements are there?

Find the number of positive integer solutions of the equation

\[x + y + z = 12.\]

Find the number of 8-digit numbers the sum whose digits is 4.

How many (positive) seven digit integers are there that have a digit sum equal to \(10\) and that are formed using only the digits \(1\), \(2\) and \(3\)?

There are 20 distinct people signed up for a raffle. There are 3 prizes that can be won: a 100 cm HDTV, a signed football jersey, and an envelope filled with gift cards. It is possible for a person to win more than one prize.

How many possible prize distributions are there?

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