7 since may be plotted by reversing the order

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Unformatted text preview: he next while the exponent on b starts at 0 and increases by one each time. Also note that the binomial coefficients themselves have a pattern. The upper number, 4, matches the exponent on the binomial (a + b)4 whereas the lower number changes from term to term and matches the exponent of b in that term. This is no coincidence and corresponds to the kind of counting we discussed earlier. If we think of obtaining (a + b)4 by multiplying (a + b)(a + b)(a + b)(a + b), our answer is the sum of all possible products with exactly four factors - some a, some b. If we wish to count, for instance, the number of ways we obtain 1 factor of b out of a total of 4 possible factors, thereby forcing the remaining 3 factors to be a, the answer is 4 . Hence, the term 4 a3 b is in the expansion. The 1 1 other terms which appear cover the remaining cases. While this discussion gives an indication as to why the theorem is true, a formal proof requires Mathematical Induction.4 To prove the Binomial Theorem, we let P (n) be the expansion formula given in the statement of the theorem...
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