The next row is shown in the animation to the right. The pattern for each row of the triangle is:
  • the left and right ends always equal 1
  • each internal element is the sum of the two elements above left and above right, like this:


If we superimpose Pascal's triangle on top of the pinball machine then we see the connection between the two: Each number of Pascal's triangle represents the number of distinct paths that a pinball can take to arrive at that point in the pinball machine:


The numbers in Pascal's triangle can also be gotten using the combination function, like this:
The combination function is defined like this:
mCn equals the number of distinct groups of n objects that can be chosen from m objects. (The ! means factorial. Note that 0! = 1. Click here for more on the factorial function.)

Thus Pascal's triangle is essentially a listing of all the possible values of the combination function.

Some examples of calculations of the combination function are:
and:


The combination function can be used for example to answer the question:
From 4 people, how many distinct groups of 2 people be chosen to work on some project? The answer is 4C2=6. If the persons are called A, B, C and D then the distinct groups are AB, AC, AD, BC, BD and CD.


In connection with the pinball game the combination function can be used to answer the question:
To arrive at a certain point, the pinball had to fall through 4 rows of pins and go right twice. How many paths lead to this point? The answer again is 4C2=6. The possible rows at which to go right are 1&2, 1&3, 1&4, 2&3, 2&4, 3&4.


In connection with algebra the combination function can be used to expand an expression like (a+b)4. Here it is:
The combination function appears here because:

For example the term a3b comes from a b from inside 1 of the brackets and a's from inside the remaining 3 brackets. There are 4C1=4 ways of choosing which bracket the b should come from. The blue arrows show one of these ways:

 

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