2.6 Pascal’s Triangle and Pascal’s Identity (Textbook Section 5.2)

Recall:  The expression can be expanded as follows:

Pascal’s Triangle:  An arrangement of numbers that are useful in binomial expansion  Binomial Expansion refers to the expansion of two terms added together, then raised to a power, in the form (a+b) n  To fill in the triangle, start with a row of 1’s on each outside diagonal  Each number is the sum of the two numbers directly above it  Begin to fill in Pascal’s Triangle – Once you have finished, work on the first 2 expansions (leave the 3 rd for later)

Properties of the Binomial Expansion of (a+b) n  The number of terms in the expansion is n+1  The first term in the expansion is a n  The last term in the expansion is b n  The degree of each term is n (since the sum of the exponents of a and b are n)  The exponents of a decrease and the exponents of b increase  The coefficients of the terms are the entries in row n of Pascal’s Triangle  Note: when numbering the rows of Pascal’s Triangle, start with the number 0 for the first row

How to find the Binomial Expansion using Pascal’s Triangle  Look at the exponent of the unexpanded term – this will tell you what row to get the coefficients from  The first term of the expansion is the “a” term of the binomial to the exponent “n”, multiplied by the 1 st coefficient in the row (this will always be 1)  The second term of the expansion will be the “a” term to the exponent “n-1” multiplied by the “b” term to the exponent “1” multiplied by the second coefficient in the row  This process continues with “a” and “b” decreasing/increasing by 1 respectively for each subsequent term until the b term is raised to the power “n” (final term in the expansion)

Pascal’s Triangle and Combinations  For large values of n, it is time consuming to expand Pascal’s triangle. An alternative method of displaying the row entries is to use combinations  In the term (a+b) n :  the exponent represents the above/top term in the combination bracket  the r-1 term represents the term in the expansion (i.e. the first term is “n choose 0”, the second term is “n choose 1”, etc.)

Pascal’s Identity  Recall: you can find an entry in Pascal's triangle by adding both entries above it  This can be expressed as Pascal’s Identity  You do not need to be able to prove THE identity, but you should be able to use it to find equivalent expressions