<p>The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is</p>
Step-by-Step Solution
Key Concept: Set up equations for digit count and sum, find all valid distributions, count arrangements for each case.
<p><strong>Solution:</strong> We need to find seven-digit numbers using digits 1, 2, 3 only such that the sum equals 10.</p><p>Let \(a\) = count of 1's, \(b\) = count of 2's, \(c\) = count of 3's.</p><p>We need: \(a + b + c = 7\) and \(a + 2b + 3c = 10\)</p><p>Subtracting: \(b + 2c = 3\)</p><p>Possible solutions: (\(a,b,c\)) = (6,1,0), (5,0,1), (4,3,0), (3,1,1), (2,0,2), (1,2,1), (0,1,2)</p><p>Number of arrangements for each case:</p><p>\(\frac{7!}{6!1!} + \frac{7!}{5!1!} + \frac{7!}{4!3!} + \frac{7!}{3!1!1!} + \frac{7!}{2!2!} + \frac{7!}{1!2!1!} + \frac{7!}{1!2!} = 7 + 42 + 35 + 140 + 630 + 1260 + 2520\)</p><p>Total = 77</p>
Correct Answer: C