Answer
$303,600$.
Work Step by Step
Apply the Fundamental Principle of Counting
If $n$ independent events occur, with $m_{1}$ ways for event 1 to occur,
$m_{2}$ ways for event 2 to occur,
$\ldots$ and $m_{n}$ ways for event $n$ to occur,
then there are $m_{1}\cdot m_{2}\cdot\cdots\cdot m_{n}$ different ways for all $n$ events to occur.
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$m_{1}=25$ ... any of the 25 boys can receive the first type of award
$m_{2}=24$ ... of the 24 remaining, one will receive the second award
$m_{3}=23$ ... of the 23 remaining, one will receive the third award,
$m_{4}=22$ ... 22 remain to choose from for the 4th award.
Total = $25\times 24\times 23\times 22$ = $303,600.$
Another approach:
Permutations, taking r=4 from n=25 boys. P(25,4)= $303,600$.
