1. **State the problem:** We want to find the number of arrangements of the digits 2, 4, 5, 5, 6, 6, and 7 that form an even number greater than 6000000. 2. **Analyze the conditions:** - The number must be 7 digits long using all digits exactly once. - The number must be even, so the last digit must be even (2, 4, or 6). - The number must be greater than 6000000, so the first digit must be 6 or 7. 3. **Digits available:** 2, 4, 5, 5, 6, 6, 7 4. **Case 1: First digit = 7** - Since 7 > 6, any number starting with 7 is > 6000000. - Last digit must be even: possible last digits are 2, 4, or 6. - We have two 6's, so last digit can be 2, 4, or 6. 5. **Count arrangements for Case 1:** - Fix first digit = 7. - Fix last digit = one of {2, 4, 6}. - Remaining digits to arrange in the middle: 5 digits from {2,4,5,5,6,6} excluding the last digit chosen. For each last digit: - If last digit = 2, remaining digits: 4, 5, 5, 6, 6 - If last digit = 4, remaining digits: 2, 5, 5, 6, 6 - If last digit = 6, remaining digits: 2, 4, 5, 5, 6 (one 6 removed) Number of arrangements for 5 digits with duplicates: - For digits with two 5's and two 6's: total permutations = \frac{5!}{2!2!} = \frac{120}{4} = 30 - For digits with two 5's and one 6: total permutations = \frac{5!}{2!} = \frac{120}{2} = 60 Calculate for each last digit: - Last digit = 2: 30 ways - Last digit = 4: 30 ways - Last digit = 6: 60 ways Total for Case 1 = 30 + 30 + 60 = 120 ways 6. **Case 2: First digit = 6** - Number must be > 6000000, so first digit = 6 is allowed. - Last digit must be even: 2, 4, or 6. - Since one 6 is used as first digit, remaining digits are 2, 4, 5, 5, 6, 7. For last digit: - Last digit = 2, remaining digits for middle: 4, 5, 5, 6, 7 - Last digit = 4, remaining digits for middle: 2, 5, 5, 6, 7 - Last digit = 6, remaining digits for middle: 2, 4, 5, 5, 7 Number of arrangements for 5 digits with duplicates (two 5's): - Total permutations = \frac{5!}{2!} = 60 For each last digit, 60 ways. Total for Case 2 = 3 * 60 = 180 ways 7. **Total number of arrangements:** $$120 + 180 = 300$$ ways **Final answer:** 300 ways