If the $(n + 1)$ numbers $a, b, c, d, . . .$ be all different and each of them a prime number, then the number of different factors (other than 1) of $a^{m} . b . c . d . . . .$ is

m - 2^{n}

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(m + 1) 2^{n}

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(m + 1) 2^{n} - 1

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None of these

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Solution

The correct option is B $(m + 1) 2^{n} - 1$
No. of elements in $b . c . d . . . = n$
Choose $a^{k}$ , where k $\in 0, 1, 2, . . m$ at a time = $(m + 1)$
for every $k$ , no. possible factors from n elements = $^{n} C_{0} +^{n} C_{1} +^{n} C_{2} + . . . . . +^{n} C_{n}$
$= 2^{n}$
Total factors $=$ No. of possible powers of $a \times$ every $k$ , no. possible factors from $n$ elements
$= (m + 1) 2^{n}$
If factor $1$ is excluded then
$(m + 1) 2^{n} - 1$

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If the (n+1) numbers a,b,c,d,... be all different and each of them a prime number, then the number of different factors (other than 1) of am.b.c.d.... is

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