If the order of numbers in a sum matters, then in how many ways can you represent the number 9 as the sum of
b
three positive integers?
Actually, sorry, I read the problem wrong. There are 28 ways.
the numbers can repeat
If the order matters, then we can look at the following: 1+8, 2+7, 3+6, 4+5. Thats 4. Since the order matters, then we could do: 8+1, 7+2, 6+3, 5+4. The total is 8. We could also use permutations as well.
Well, if the order matters, then we'll have to get creative with our jokes too! Let's see... representing the number 9 as the sum of three positive integers? That sounds like a math problem designed to make me go bananas! 🍌
Now let's think about this "fruitful" situation. We could have two big bananas and one little banana, like 6 + 2 + 1 = 9. Or we could have one big banana and two little bananas, like 4 + 3 + 2 = 9. But wait, there's more! We could even have three medium-sized bananas, like 3 + 3 + 3 = 9.
So, if we let our imagination run "a-peel", we can come up with different combinations to represent the number 9 as the sum of three positive integers: 6 + 2 + 1, 4 + 3 + 2, and 3 + 3 + 3. That makes a total of three ways! 🍌🍌🍌
But hey, I'm just a silly Clown Bot, so take these "fruitful" calculations with a grain of salt! 😄
To find the number of ways to represent the number 9 as the sum of three positive integers, where the order matters, we can use a technique called "stars and bars" or "balls and urns".
In this case, let's imagine we have 9 stars that we need to divide into three groups. We can use two bars to separate the stars into three groups.
For example, if we have 9 stars and 2 bars arranged like this: ★★|★|★★★★, it represents the sum 2 + 1 + 6 = 9.
Since there are 9 stars and 2 bars, we have a total of 11 objects (stars and bars) to arrange. We can think of placing the two bars in the 11 positions (each position represents the space between two objects or at the beginning/end), and the stars will fill in the remaining positions.
To find the number of ways to arrange these objects, we need to choose the positions for the bars. We can choose 2 out of the 11 positions using combinations, which can be calculated as C(11, 2).
Therefore, the number of ways to represent the number 9 as the sum of three positive integers, where the order matters, is C(11, 2).
Using the formula for combinations, C(n, r) = n! / (r!(n-r)!), we can calculate:
C(11, 2) = 11! / (2! * (11-2)!) = (11 * 10) / (2 * 1) = 55.
So there are 55 ways to represent the number 9 as the sum of three positive integers, where the order matters.
Lets just look at this combo
1+2+6
1+6+2
2+1+6
2+6+1
6+1+2
6+2+1
or for that combination, six ways 3*2
Now note that you can't have a seven or 8 and get three numbers to add to nine.
1+3+5 combinations, again six ways
2+3+4 combinations, again six ways
so I count 18 ways