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Luke of Madras College and Ling Xiang Ning, Raffles Institution, Singapore, sent us answers to the question "How many divisors does $n!$ have?"
Luke gave more detail:
To understand the reason for this, think of each divisor being expressed as a product of prime factors. The number of possible divisors is found by working out the number of ways of choosing the prime factors of the divisor. The prime $p_1$ may not occur as a factor of the divisor, or it may occur to the power 1 or 2 or 3 or any power up to at most $a$, hence there are $(a+1)$ possibilities for a divisor to contain $p_1$ as a factor. Similarly there are $(b+1)$ possibilities for a divisor to contain $p_2$ as a factor and $(c+1)$ possibilities for a divisor to contain $p_3$ as a factor and so on. We multiply these numbers of possibilities to find the total number of possibilities.