<p>How many integers > 100 and < 10<sup>6</sup> have the digital sum = 5?</p>
<p style="display:inline">258</p>
<p style="display:inline">246</p>
<p style="display:inline">248</p>
<p style="display:inline">252</p>
Step-by-Step Solution
Key Concept: Apply the stars and bars formula for non-negative integer solutions to a linear equation where variables represent the digits of the numbers up to the maximum power of ten.
<p>Let abcdef represent a number less than 10<sup>6</sup>. Consider,<br />
<span class="math-tex">$\left.\begin{array}{l} a+b+c+d+e+f=5 \\ a, b, c, d, e, f(a l l) \geq 0 \end{array}\right\}$</span> ...(i)<br />
Observe that a = 0 corresponds to four digit numbers, a and b = 0 correspond to three digit numbers, and so on<br />
Number of non negative integer solutions of (i)<br />
= <sup>10</sup>C<sub>5</sub> = <span class="math-tex">$\frac{10 \times 9 \times 8 \times 7 \times 6}{1 \times 2 \times 3 \times 4 \times 5}$</span> = 252<br />
This number includes the number of 1 digit, 2 digit numbers having digital sum 5. (5, 14, 41, 23, 32, 50 are 6 numbers that have been counted.)<br />
Required number = 252 - 6 = 246</p>
Correct Answer: B