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Daily Magic Spell Discussion - February 10, 2017

Daily Magic Spell Discussion - February 10, 2017

Today, we will discuss about the Daily Magic Spell proposed on February 10, 2017.

Problem: Suppose you have an $8$-digit number. How many such numbers have the sum of their digits being $10$?

Solution: Suppose that $abcdefgh$ is a $8$ digit number. Then we want $a+b+c+d+e+f+g+h = 10$. Since $a \geq 1$, we must put one ball in the first box, leaving $9$ balls to distribute in $8$ boxes.

$\displaystyle \binom{9+8-1}{9} = 11440$

However, we cannot place all $10$ balls in the first bin since this would indicate that the first digit is $10$. Therefore, the number of $8$ digit numbers with digit sum $10$ is

$11440 - 1 = 11439$.

This question is pretty difficult for various of reasons. The most common mistake in solving this problem is failing to account for the case when one places all $10$ balls in the first digit. Since $10$ doesn't represent a one-digit number, we need to remove this case.

Additionally, the problem is posed as a number theoretic problem, so one may try to implement a method using their knowledge in number theory. This may mislead students to yield an incorrect response from the incorrect methods.

If one successfully realizes that this problem is a counting problem, it is very likely that student's would attempt to consider casework. Specifically, they would consider the case when the first digit is $1$, then $2$, and so on. Although, it is possible to obtain the correct answer this way, it quickly becomes tedious and the approach is extremely prone to errors.

If you have any other approaches that you believe is worth discussing about, please let us know by posting here! We would greatly appreciate listening to your ideas!