Question

The number of numbers between \(345\) and \(543\) such that the sum of the digits in each number is \(15\) is

• A. \(24\)
• B. \(136\)
• C. \(91\)
• D. \(17\)

Hint:

For digit-sum problems, fix the hundreds digit first and form an equation for the remaining digits. \[ \boxed{\text{Hundreds digit}+\text{Tens digit}+\text{Units digit}=\text{Required sum}} \] Then count only those pairs satisfying the given range.

Answer 1

The Correct Option is D

Step 1: Consider numbers from \(345\) to \(399\). Let the number be \[ 3xy. \] Since the sum of digits is \(15\), \[ 3+x+y=15, \] or \[ x+y=12. \] Also, the number must be at least \(345\). The possible pairs are \[ (4,8),(5,7),(6,6),(7,5),(8,4),(9,3). \] Hence, there are \[ \boxed{6} \] such numbers.

Step 2: Consider numbers from \(400\) to \(499\). Let the number be \[ 4xy. \] Then, \[ 4+x+y=15, \] or \[ x+y=11. \] The possible pairs are \[ (2,9),(3,8),(4,7),(5,6),(6,5),(7,4),(8,3),(9,2). \] Thus, the number of such numbers is \[ \boxed{8.} \]

Step 3: Consider numbers from \(500\) to \(543\). Let the number be \[ 5xy. \] Then, \[ 5+x+y=15, \] or \[ x+y=10. \] Since the number does not exceed \(543\), \[ x\le4. \] The possible pairs are \[ (1,9),(2,8),(3,7). \] Thus, there are \[ \boxed{3} \] such numbers.

Step 4: Find the total count. Hence, \[ 6+8+3=17. \] Therefore, \[ \boxed{17} \] is the required number. Hence, \[ \boxed{(D)} \] is the correct answer.