Question

Three digit numbers in which the middle one is a perfect square are formed using the digits $1$ to $9$. Their sum is

• A. $134055$
• B. $270540$
• C. $170055$
• D. $270504$

Answer 1

Answer: (A)

$134055$

Let $ABC$ be the number

$Aϵ\{1,2,\mathrm{3.....9}\}$ $Bϵ\{1,4,9\}$

$Cϵ\{0,1,2,.....9\}$

Each possible choice for $B$ occurs. $81$ times $[9\times 9]$

Each possible choice for $A$ & $C$ occurs $27$ times $[3\times 9]$

$\therefore$ Sum of all numbers

$=27\times 100\times (1+2+3+.....+9)+81\times 10(1+4+9)+27\times 1\times (1+2+\mathrm{3.....9})$

$=27\times 45\times 100+81\times 10\times 14+27\times 1\times 45$

$=134055$

Hence, this is the answer.