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Unformatted text preview: Section 11.5 Binomial Theorem The Binomial Theorem gives a formula for the expansion of ( x + a ) n where n is a natural number. The coefficients in the expansion of ( x + a ) n can be easily found in a number of ways, including the the use of Pascals Triangle and the use of a combinations formula from Probability Theory. We begin by writing the terms of a few of the binomial expansions: ( x + a ) 1 = x + a ( x + a ) 2 = x 2 + 2 ax + a 2 ( x + a ) 3 = x 3 + 3 ax 2 + 3 a 2 x + a 3 Pascals Triangle The coefficients of the terms in the binomial expansions for ( x + a ) n can be obtained from the numbers in Pascals Triangle which is shown in the following. The symbol n C j = n j is used in probability theory and the values of n C j give the coefficients of the terms in binomial expansions. The value of n C j is defined as follows: n C j = n j = n ! j !( n- j )! Note. In Probability Theory, n C j is the number of different combinations of n items taken j at a time. For example,time....
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