Factors of a prime‑power product

Let $a,b,c,d,\dots$ be $n+1$ distinct prime numbers and let $m$ be a positive integer. Prove that the number of distinct positive divisors of the integer $a^m{b}^{c}d\dots$ is $(m+1)2^n-1$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XI. Permutations and Combinations, Examples XI.b, No. 33