Number of Non-Negative Integral Solutions for a + b + c + d + e + f = 17

To find the number of non-negative integral solutions for the equation a + b + c + d + e + f = 17, we can use a combinatorial approach known as stars and bars or balls and urns.

Stars and Bars Method

The stars and bars method is a combinatorial technique used to count the number of ways to distribute identical items into different groups.

Step 1: Transform the Equation

To use the stars and bars method, we need to transform the given equation into a form suitable for application.

Let's denote the variables a, b, c, d, e, and f as non-negative integers.

a + b + c + d + e + f = 17

Step 2: Introduce Bars

We introduce 5 bars to separate the variables a, b, c, d, e, and f. These bars divide the total sum of 17 into 6 groups.

For example, if we represent a solution as:

**|***|****|**|*|***|*

This representation implies that a = 0, b = 3, c = 4, d = 0, e = 1, and f = 9.

Step 3: Counting the Solutions

To count the number of non-negative integral solutions, we need to find the number of ways to arrange the 17 stars and 5 bars.

In this case, we have a total of 22 objects (17 stars and 5 bars). The number of ways to arrange these objects can be calculated using the concept of permutations.

The formula to calculate permutations is given by:

P(n, r) = n! / (n - r)!

Where n is the total number of objects and r is the number of identical objects.

In our case, we have:

P(22, 5) = 22! / (22 - 5)!

Simplifying this expression, we find:

P(22, 5) = 22! / 17!

Step 4: Calculate the Solution

Evaluating the above expression, we get:

P(22, 5) = (22 * 21 * 20 * 19 * 18 * 17!) / 17!

Simplifying further, we find:

P(22, 5) = 22 * 21 * 20 * 19 * 18

Calculating this expression, we get:

P(22, 5) = 3,818,760

Therefore, there are 3,818,760 non-negative integral solutions for the equation a + b + c + d + e + f = 17.

Nhi aaya yaaar