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| Alternative Proof of Binomial Theorem for Positive Integral Index (Combinatorial Method) |
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| We have, (a + b)n = (a + b) (a + b) ....... n times. |
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| The terms on the RHS are obtained by taking one letter from each factor and multiplying them together. |
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| Choosing 'a' from all the factors, we get the term an. |
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| Choosing 'a' from (n-1) factors and 'b' from the remaining factor, we get an-1b. The factor for 'b' can be chosen in nC1 ways. Thus, there are nC1 terms each equal to an-1b. Choosing 'a' from (n-2) factors and 'b' from the remaining two factors, we get an-2b2. The two factors for 'b' can be chosen in nC2 ways. Thus, there are nC2 terms each equal to an-2b2. |
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| Proceeding in this way, we have nCk terms each equal to an-kbk. |
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| Finally choosing 'b' from each factor, we get the term bn. |
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