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Pascal's Triangle

Let us see the expansions of for values of n = 0, 1, 2, 3, 4 etc and study their pattern.

We observe:
  1. The number of terms in each of the expansions in one more than the power to which a + b is raised
  2. The power (index) of the first quantity in the binomial starts with the power of the binomial (a + b) and goes on decreasing one per term and the that of the second quantity keeps increasing one at a time for each term till it reaches the maximum power in the last term of the expansion
  3. The total power (index) of each term (i.e. index of a + index of b ) in each term is equal to the index of a + b.
Using these observations, we can form a table of coefficients of the above binomial's powers

As you can see the sum of the two quantities on the top of the triangle (inverted) is the quantity at the bottom (tip) of the triangle. This will help you to find the expansion of

We can go on till any index we want and this is exactly Pascal's triangle.

Pascal's Triangle for n = 12, is given below:

Note: Please recollect which you have already done in the “Permutations and Combinations” Chapter.

Example: Using Pascal's triangle, write down the expansion of


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