<p>If the number of 6 digit natural numbers such that sum of their digits is 10 and digits 0, 1, 2 and 3 occur at least once in them is k then find k/100</p>
Step-by-Step Solution
Key Concept: We need to find 6-digit numbers where digits 0, 1, 2, 3 each appear at least once, sum of all digits equals 10, and the first digit cannot be 0. Use inclusion-exclusion with stars and bars after fixing the mandatory digits.
<p><strong>Step 1:</strong> Since digits 0, 1, 2, 3 must occur at least once, we place one of each: sum = 0 + 1 + 2 + 3 = 6. We need remaining 2 positions (since 6 - 4 = 2) to sum to 10 - 6 = 4.</p><p><strong>Step 2:</strong> We need to distribute 4 among 2 remaining digits. Let these digits be d₅ and d₆, where each dᵢ ∈ {0,1,2,...,9}. We solve: d₅ + d₆ = 4.</p><p><strong>Step 3:</strong> Possible solutions for (d₅, d₆) are: (0,4), (1,3), (2,2), (3,1), (4,0). This gives 5 cases.</p><p><strong>Step 4:</strong> For each case, we have a multiset of 6 digits. Count distinct arrangements using the formula for permutations of multisets.</p><p><strong>Case 1: Digits are {0,0,1,2,3,4}</strong> - Two 0's. Arrangements = 6!/2! = 360. First digit cannot be 0: arrangements starting with 0 = 5!/2! = 60. Valid = 360 - 60 = 300.</p><p><strong>Case 2: Digits are {0,1,1,2,3,3}</strong> - Two 1's and two 3's. Arrangements = 6!/(2!×2!) = 180. Starting with 0 = 5!/(2!×2!) = 30. Valid = 180 - 30 = 150.</p><p><strong>Case 3: Digits are {0,1,2,2,3,2}</strong> - Three 2's. Arrangements = 6!/3! = 120. Starting with 0 = 5!/3! = 20. Valid = 120 - 20 = 100.</p><p><strong>Case 4: Digits are {0,1,2,3,3,1}</strong> - Two 1's and two 3's. Arrangements = 6!/(2!×2!) = 180. Starting with 0 = 5!/(2!×2!) = 30. Valid = 180 - 30 = 150.</p><p><strong>Case 5: Digits are {0,1,2,3,4,0}</strong> - Two 0's. Arrangements = 6!/2! = 360. Starting with 0 = 5!/2! = 60. Valid = 360 - 60 = 300.</p><p><strong>Step 5:</strong> Total k = 300 + 150 + 100 + 150 + 300 = 1000.</p><p><strong>Step 6:</strong> k/100 = 1000/100 = 10.</p><p><strong>∴ Answer: 10</strong></p>
Correct Answer: 10