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Intermediate Mathematics - Binomial Theorem - Pascal's Triangle
Binomial Theorem - Pascal's Triangle
Observe the pattern in the following binomial expansions. You should
verify these by writing down the factors and multiplying them out. The
number of factors is given by n:
* The first term in each expansion is a and the last term is b.
* As we work from left to right, the power of a decreases by 1 while the
power of b increases by 1.
* Each term in the expansion has degree n. This means that the powers of
a and b in each term add to n.
* There are n+1 terms in each expansion.
The coeffficients in each expansion are known as the binomial coefficients
and they form the following pattern known as Pascal's Triangle:
and so on. Each row begins and ends with 1. Each other number can be
obtained by adding the two numbers in the line immediately above to the
right and left. For example, in line four each of the 3's is obtained by
adding 1+2 from immediately above. In line five we have 4=1+3, 6=3+3,
4=1+3. The rows are symmetrical.
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