(1, 1, 8) is a triple of natural numbers which has a sum of 10. Consider (1, 8, 1) and (8, 1, 1) to be the same triple as (1, 1, 8). How many different triples of natural numbers have a sum of 10? Include (1, 1, 8) as one of your triples.
Is there any quicker way?
No there is not a quicker way unless you want to use a computer code to cheat/get a hint.
To find the number of different triples of natural numbers that have a sum of 10, we can break down the problem step by step:
Step 1: Determine the range of possible values for each number in the triple. Since we are dealing with natural numbers (positive integers), the smallest number in the triple can be 1, and the largest number in the triple cannot exceed 10.
Step 2: Find all possible combinations of three natural numbers that sum up to 10.
We can approach this by listing out all the possibilities. Let's start with the largest number, 10:
(10, ?, ?)
Since the sum of the triple must be 10, we need to find two numbers that sum up to 10 when added to 10. The only possibility is (10, 1, 1):
(10, 1, 1)
Moving on to the next largest number, 9:
(9, ?, ?)
Similarly, we need to find two numbers that sum up to 9 when added to 9. The possibilities are (9, 1, 1) and (9, 2, 1):
(9, 1, 1)
(9, 2, 1)
Continuing this process, we can find all the possible triples that have a sum of 10:
(8, 1, 1)
(8, 2, 1)
(8, 3, 1)
(8, 4, 1)
(8, 5, 1)
(8, 6, 1)
(8, 7, 1)
(8, 8, 1)
(8, 2, 2)
(8, 3, 2)
(8, 4, 2)
(8, 5, 2)
.
.
.
(3, 6, 1)
(3, 7, 1)
(3, 8, 1)
(3, 4, 3)
(3, 5, 2)
(3, 6, 2)
(3, 7, 2)
(3, 4, 4)
(3, 5, 3)
(3, 6, 3)
And so on, until we reach the smallest number, 1:
(1, 1, 8)
(1, 2, 7)
(1, 3, 6)
(1, 4, 5)
(1, 5, 4)
(1, 6, 3)
(1, 7, 2)
(1, 8, 1)
Step 3: Count the number of unique triples we have found. From the listing above, we can see that there are 28 different triples of natural numbers that have a sum of 10, including (1, 1, 8) as one of them.
Therefore, the answer is 28.
since order does not matter, start small and work your way up.
1,1,8
1,2,7
1,3,6
...
3,3,4