The Multinomial Coefficient

                                                        The Multinomial Coefficient 

Defining  it looks like

  

 but it has an extra set of parentheses around and  it is the number of ways to choose k         objects from a set of n objects, where order is not important, but repetition is allowed.

The equation is

Lets try a Couple of examples using this formula:

Example One: Find the number of solutions to the equation a + b + c = 40, where a, b, and c are all non-negative integers.

Example Two:  Suppose you have 15 ice cream flavors and you want a cup of five scoops, they can be the same flavor. How many ways can we choose the flavors?

Example Three:  How many ways can we arrange the letters Richelle?

Well we see that there are two L’s and two E’s one R, one I, one C and one H. So we can use the other equation, which introduces the multinomial coefficient, the equation is

There are eight letters in Richelle. So there are

 The Right arrow is point to the multinomial coefficient.

So the answer to how many ways we can arrange Richelle is

Example Four: How many ways can you arrange the letters in Halloween?

There is 1 H, 1 A, 2 L’s, 1 O, 1 W, 2 E’s and 1 N. in Halloween.

The Multinomial Theorem states:

Where a and b are non-negative numbers that sum up to n. We will use this theorem later.

Test Question: How many ways can you arrange the letters in Nevada State College, you can count them as all one word?

Test Question: Find the number of solutions to the equation a + b + c + d +e + f =150, where a,b,c.d.e and f are all non-negative integers?