Question

The number of all 3-digit numbers abc (in base 10) for which 

(a × b × c) + (a × b) + (b × c) + (c × a) + a + b + c = 29 is 

 (A) 6 

(B) 10 

(C) 14 

(D) 18 

Answer 1

Correct option (C) 14

Explanation:

abc + ab + bc + ca + a + b + c = 29 

⇒ abc + ab + bc + ca + a + b + c + 1 

= 30 ⇒ (a + 1) (b + 1) (c + 1) = 30  

Product can be 2 × 3 × 5 ⇒ digits are 1, 2, 4 so possible number = 3! = 6 

 Product can be 1 × 5 × 6 ⇒ digits are 0, 4, 5 so possible number = 2 × 2! = 4 

 Product can be 1 × 3 × 10 ⇒ digits are 0, 2, 9 so possible number = 2 × 2! = 4 

 Total numbers = 14