- In each expansion there are
*n*+ 1 terms. (For example the bottom (*n*= 5) expansion has 6 terms.)

- The powers of
*x*start at*n*and decrease by 1 in each term until they reach 0.

- The powers of
*y*start at 0 and*increase*by 1 until they reach*n*.

- The coefficients in each expansion add up to 2
^{n}. (For example in the bottom (*n*= 5) expansion the coefficients 1, 5, 10, 10, 5 and 1 add up to 2^{5}= 32.)

- The coefficients are exactly the numbers that appear in
**Pascal’s Triangle**, shown here: What is Pascal’s Triangle? It is an infinitely tall triangle of numbers that can be constructed as follows:

- The first number and the last number in each row is a 1.

- Every number in the interior of the triangle is the sum of the two numbers above left and above right, like this:

- Pascal’s Triangle above is animated to show how each row is generated from the row above.

- Each number in Pascal’s triangle can also be generated using the so-called
**combination function**as shown here:**Pascal’s triangle is really just a listing of all the possible values of the combination function.**The combination function is defined as where*n*! or “*n*factorial” is The combination function gives the number of distinct groups of*k*objects that can be chosen from*n*distinct objects when the order in which they are chosen doesn’t matter.

Here are some examples of calculations that use the combination function:

**Example:**From 4 people, how many distinct groups of 2 people be chosen to work on some project?

**Solution:**The answer is If the persons are called A, B, C and D then the distinct groups are AB, AC, AD, BC, BD and CD.

**Example:**In the expansion of (*a*+*b*)^{4}what is the coefficient of the term*a*^{3}*b*?

**Solution:**The answer is 4. The reason is that the term*a*^{3}*b*comes from multiplying a*b*from inside 1 of the brackets and*a*’s from inside the remaining 3 brackets. There are ways of choosing which bracket the*b*should come from. The blue arrows show one of these ways:

Putting all of this together gives the**binomial theorem**, which states that a power of a binomial can be expanded like this: Using summation notation the binomial theorem can also be written like this: - The first number and the last number in each row is a 1.

Consider (

(then apply the distributive law, and then simplify by collecting like terms. After distributing, but before collecting like terms, there are 81 terms. (This is because every term in the first brackets has to be multiplied by every term in the second brackets, giving 9 terms. Each of these has to be multiplied by every term in the third brackets, giving 27 terms. Finally each of these has to be multiplied by every term in the fourth brackets, giving 81 terms.) Many of the terms look different before simplifying, but are identical after simplifying. For example the four termsa+b+c) (a+b+c) (a+b+c) (a+b+c),

- There are 15 distinct terms.

- Each term is of degree 4.

- The coefficients add up to 3
^{4}= 81 (As mentioned above this is the number of terms before collecting like terms.)

- Start with three nested summations:

The problem is that this triple summation produces 5·5·5=125 terms of various degrees ranging from*a*^{0}*b*^{0}*c*^{0}to*a*^{4}*b*^{4}*c*^{4}.

- Introduce a
*filter factor*that equals 1 for the wanted terms (those of degree 4) and 0 for the unwanted terms. This is the**Kronecker delta function**, denoted*δ*and defined as: Now change the summation to read It contains only the 15 wanted terms of degree 4. The only remaining problem is that every coefficient equals 1._{i, j}

- Construct the correct coefficient. This is the so-called
**multinomial coefficient**: This multinomial coefficient gives the number of ways of depositing 4 distinct objects into 3 distinct groups, with*i*objects in the first group,*j*objects in the second group and*k*objects in the third group, when the order in which they are deposited doesn’t matter.

For example the coefficient of the*a*^{1}*b*^{1}*c*^{2}term uses*i*= 1,*j*= 1 and*k*= 2 and equals With this coefficient the expansion reads

We can generalize this to give us the

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