There is a young boy's birthday party which 3 friends have attended. The mother has arranged 10 games where a prizeis awarded for winning a game. The prizes are identical. If each of the 4 children receives at least one prize, then howmany distributions of prizes are possible?
Step 1: Understand the problem statement.
Total number of identical prizes () = (since there are games and each game awards one identical prize).
Total number of children () = birthday boy + friends = children.
Condition: Each child must receive at least one prize.
Step 2: Use the Stars and Bars theorem.
We need to find the number of ways to distribute identical objects among distinct individuals such that each individual receives at least object.
According to the Stars and Bars method (for positive integer solutions), the number of ways to distribute identical items into distinct groups where each group gets at least one item is given by the formula:
Step 3: Substitute the values into the formula.
Here, and :
Step 4: Calculate the combination.
Therefore, there are possible ways to distribute the prizes.
Correct Option: B (84)