Question

The number of factors (excluding $1$ and the expression itself) of the product of $a^7{b}^4{c}^{3}def$ where $a,b,c,d,e,f$ are all prime numbers is

• A. $1278$
• B. $1360$
• C. $1100$
• D. $1005$

Answer 1

Answer: (A)

$1278$

A factor of expression $a^7{b}^4{c}^{3}def$ is simply the result of selecting one or more letters from $7a^{\prime }s,4b^{\prime }s,3a^{\prime }s,d,e,f.$
The collection of letters can be observed as a collection of $17$ objects out of which $7$ are alike of one kind (a's), $4$ are of second kind (b's), $3$ are of third kind (c' s) and $3$ are different $(d,e,f)$
The number of selections =$(1+7)(1+4)(1+3)(1+1{)}^3=8\times 5\times 4\times 8=1280.$
But we have to exclude two cases : (i) When no letter is selected (ii) When all are selected.

So, the number of factors $=1280-2=1278$
Hence, option ${}^{\prime }{A}^{\prime }$ is correct.