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| Method of writing expansion for (a+b)n |
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| The first term in the expansion of (a + b)n is nC0anb0. For the second term, the coefficient is taken as nC1, the power of 'a' is decreased by one and the power of 'b' is increased by one. So, the second term is nC1an-1b1. For the third term, the coefficient is taken as nC2, the power of 'a' is again decreased by one and the power of 'b' is increased by one. This process goes on, till we get the last term as nCna0bn. |
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| Some Observations |
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For n  N, in the expansion of (a + b)n, we observe that: |
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 the number of terms is n+1 |
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 the exponent of 'a' decreases from n to 0 |
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 the exponent of 'b' increases from 0 to n |
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 the sum of exponents of 'a' and 'b' in any term is n. |
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 the coefficient of any term is nCk where k is the exponent of 'b'. |
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 nC0, nC1, nC2, ..... nCn are called the binomial coefficients. |
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| Since, nCr = nCn-r, we have nC0 = nCn, nC1 = nCn-1, nC2 = nCn-2, .... |
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 The binomial coefficients in (a + b)n, which are equidistant from beginning and end are equal. |
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| The binomial coefficients in the expansion of (a + b)n can be easily evaluated by using the Pascal's triangle given below: |
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| In the Pascal's triangle, each row starts and ends with 1 and each coefficient in a row is equal to the sum of the two coefficients one just before it and other just after it in the preceding row. |
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