The number of ways by which 6 distinct balls can be put in 5 distinct boxes are
[amp_mcq option1=”7776″ option2=”15625″ option3=”720″ option4=”120″ correct=”option2″]
This question was previously asked in
UPSC CAPF – 2020
Consider the balls one by one:
The first ball can be put into any of the 5 distinct boxes (5 choices).
The second ball can be put into any of the 5 distinct boxes (5 choices).
…
The sixth ball can be put into any of the 5 distinct boxes (5 choices).
Since the choice for each ball is independent, the total number of ways is the product of the number of choices for each ball.
Total ways = $5 \times 5 \times 5 \times 5 \times 5 \times 5 = 5^6$.
$5^6 = 15625$.
This is the general formula for placing $n$ distinct items into $m$ distinct bins: $m^n$. Here $n=6$ and $m=5$, so $5^6$.