Grades 11-12 Video Solutions 2023-2023_11-12

Video Transcription

Problem number 12. For positive integer n, n factorial is defined as a product of all integers  from 1 to n. For example, 4 factorial equals 4 times 3 times 2 times 1 equals 24. What is the  sum of the digits of n if n factorial equals 6 factorial times 7 factorial? Alright, so for this  problem we rewrite n factorial as 7 factorial times 6 factorial, so that's equal to 7 factorial  times 6 times 5 times 4 times 3 times 2 factorial, which is just equal to times 2. Now what we're  going to try to do is try to get the next numbers after 7 factorial from 6 times 5 times 4 times 3  times 2, so that I can get to some n factorial number and then we can sum the digits. So we set  7 factorial times, we rewrite the order using the commutative property of multiplication, so times  4 times 2 times 3 times 6 plus 5. Now what we get is 7 factorial times 4 times 2, put that in  parentheses, times 3 times 3, notice we split the 6 into 3 times 2, and then times 2 times 5, so we  get 7 factorial times 8 times 9 times 10, hey wait a second that's just 10 factorial, so the correct  answer would be A. Now one note about this problem, you don't actually need to do this whole computation  out, you can use some tricks. You know that the answer has to be less than 11 because if n is 11  or greater, then n factorial would have a factor of 11, but the right side does not have a factor  of 11, so it has to be less than 11. Now there's two factors of 5 on the right side because 6  factorial has a factor of 5 and 7 factorial has a factor of 5, and so n factorial must have two  factors of 5, so n must be greater than 10, greater than or equal to 10. So n must be greater than or  equal to 10, so that has two factors of 5, and n must be greater than 11 or less than 11, so we have  zero factors of 11. The only number that satisfies both conditions is n equals 10, and so our answer would be 1 plus 0 equals A.

Video Summary

The problem involves finding the sum of the digits of \( n \) when \( n! = 6! \times 7! \). By breaking down the factorials and rearranging them, the solution identifies that \( n! = 10! \). Since \( n = 10 \), the sum of its digits, \( 1 + 0 \), equals 1. The final answer is 1.