Question:

Step 1: Determine the total number of outcomes. There are 5 different games and 4 children. Each game can be distributed to any of the 4 children independently. The total number of ways to distribute the games is $4^5$. $$ \text{Total Outcomes} = 4^5 = 1024 $$ Step 2: Determine the number of favorable outcomes. A favorable outcome is one where each child receives at least one game. This is equivalent to finding the number of surjective functions from a set of 5 games to a set of 4 children. We use the Principle of Inclusion-Exclusion. The number of surjective functions from a set of $m$ elements to a set of $n$ elements is given by: $$ \sum_{k=0}^{n} (-1)^k \binom{n}{k} (n-k)^m $$ In this case, $m=5$ and $n=4$: $$ \text{Favorable Outcomes} = \binom{4}{0}(4-0)^5 - \binom{4}{1}(4-1)^5 + \binom{4}{2}(4-2)^5 - \binom{4}{3}(4-3)^5 + \binom{4}{4}(4-4)^5 $$ $$ = 1 \cdot 4^5 - 4 \cdot 3^5 + 6 \cdot 2^5 - 4 \cdot 1^5 + 1 \cdot 0^5 $$ $$ = 1 \cdot 1024 - 4 \cdot 243 + 6 \cdot 32 - 4 \cdot 1 + 0 $$ $$ = 1024 - 972 + 192 - 4 $$ $$ = 240 $$ Step 3: Calculate the probability $p$. The probability $p$ is the ratio of favorable outcomes to the total number of outcomes. $$ p = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{240}{1024} $$ To simplify the fraction, divide both numerator and denominator by their greatest common divisor. $$ p = \frac{240 \div 16}{1024 \div 16} = \frac{15}{64} $$ Step 4: Calculate $1/p$. $$ \frac{1}{p} = \frac{1}{15/64} = \frac{64}{15} $$ Step 5: Apply the greatest integer function. The greatest integer function $[\cdot]$ gives the largest integer less than or equal to the input. $$ \left[\frac{1}{p}\right] = \left[\frac{64}{15}\right] $$ To evaluate this, we perform the division: $$ \frac{64}{15} = 4 \frac{4}{15} \approx 4.2666... $$ Therefore, $$ \left[\frac{64}{15}\right] = [4.2666...] = 4 $$