How many ordered triples (a, b, c) of positive integers are there which satisfy the equation a + b + c = 10 ?
if a=1, b+c=9, so 8 there
if a=2, b+c=8. so 7 there
...
if a=8, b+c=2, so 1 there
Looks like 1+2+...+8 = 36
If no duplicates are allowed, the number goes way down.
hi everyone :(
To find the number of ordered triples (a, b, c) of positive integers that satisfy the equation a + b + c = 10, we can use the concept of generating functions.
Let's consider the exponential generating function for the sum a + b + c:
G(x) = (x + x^2 + x^3 + ...)3 = x^3 / (1 - x)^3
The coefficient of x^10 in this generating function will give us the number of solutions to a + b + c = 10.
To find the coefficient, we can use the concept of the binomial series expansion:
(1 - x)^(-3) = 1 + 3x + 6x^2 + ...
Therefore, the coefficient of x^10 is the coefficient of x^10 in the expansion of (x^3)(1 + 3x + 6x^2 + ...).
The term with x^10 can only be obtained by multiplying x^3 with x^7 from the expansion of (1 + 3x + 6x^2 + ...):
Coefficient of x^10 = (1)(6) = 6
Hence, there are 6 ordered triples (a, b, c) of positive integers that satisfy the equation a + b + c = 10.
To find the number of ordered triples (a, b, c) of positive integers that satisfy the equation a + b + c = 10, we can use a combinatorial method called stars and bars.
The stars and bars method involves visualizing the problem using stars (*) and bars (|). In this case, we have 10 stars representing the sum of a, b, and c, and we need to separate them into three groups (a, b, c) using two bars.
For example, if we have 10 stars and 2 bars:
**|***|****
The stars to the left of the first bar represent the value of a, the stars between the two bars represent the value of b, and the stars to the right of the second bar represent the value of c.
Since each group (a, b, c) must have at least one star, we initially place one star in each group to meet the positive integer requirement:
*|*|*
Now, we have 7 stars remaining, which we need to distribute among the three groups. This is equivalent to arranging 7 identical stars and 2 identical bars. Using the stars and bars formula, the number of ways to arrange these objects is given by the binomial coefficient, which can be calculated as:
(7 + 2 - 1) choose (2) = 8 choose 2 = 28
Therefore, there are 28 ordered triples (a, b, c) of positive integers that satisfy the equation a + b + c = 10.