There is a young boy's birthday party which 3 friends have attended. The mother has arranged 10 games where a prize is awarded for winning a game. The prizes are identical. If each of the 4 children receives at least one prize, then how many distributions of prizes are possible?
Step-by-step Solution:
Total number of prizes: 10 \[\] Number of recipients: 10 \[\] The number of ways is equal to the number of solutions of the equation: \[ x + y + z + w = 10 \] where \( x, y, z, w \geq 1 \). Hence, the number of solutions is: \[ \binom{10-1}{4-1} = \binom{9}{3} = 84 \]