Seven from Nine
In how many ways can seven of the numbers 1-9 be chosen such that they add up to a multiple of 3?
Age
11 to 14
Challenge level
Problem
In how many different ways can seven different numbers be chosen from the numbers $1$ to $9$ so that the seven numbers have a total which is a multiple of $3$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 12 ways
Choosing 7 numbers is the same as choosing which 2 numbers to leave out
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
= (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5 = 45
45 is a multiple of 3
$\therefore$ the two left out add up to a multiple of 3
3,6 3,9 6,9 are the ways of pairing two multiples of 3
1,2 1,5 1,8 are the ways of pairing 1 with a number 1 less than a multiple of 3
4,2 4,5 4,8
7,2 7,5 7,8 are the ways of pairing the other numbers which are 1 more than
a multiple of 3 with a number 1 less than a multiple of 3