(A) To find the highest power of a prime $p$ in $n!$,we use Legendre's formula: $E_p(n!) = \sum_{k=1}^{\infty} \lfloor \frac{n}{p^k} \rfloor$.Here,$n

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(A) To find the highest power of a prime $p$ in $n!$,we use Legendre's formula: $E_p(n!) = \sum_{k=1}^{\infty} \lfloor \frac{n}{p^k} \rfloor$.Here,$n = 100$ and $p = 3$.$E_3(100!) = \lfloor \frac{100}{3} \rfloor + \lfloor \frac{100}{9} \rfloor + \lfloor \frac{100}{27} \rfloor + \lfloor \frac{100}{81} \rfloor$.$E_3(100!) = 33 + 11 + 3 + 1 = 48$.

Class 11 Mathematics Permutations and Combinations Multinomial theorem, Number of divisors

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What is the highest power of $3$ in $100!$?

A
$48$
B
$44$
C
$45$
D
$33$

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Permutations and Combinations

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Multinomial theorem, Number of divisors

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What is the highest power of $3$ in $100!$?

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