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| Some Applications of Binomial Theorem for Positive Integral Index |
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| Using Binomial theorem, prove that: |
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| When x = 1, |
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| When x = -1 |
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| \ Hence (i) is proved. |
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| \ Hence (ii) is proved. |
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| \ Hence (iii) is proved. |
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| Note 1: |
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| The L.H.S of (ii) and (iii) will have only finite number of terms, because nCr = 0 in case r>n. |
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| Note 2: |
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| nC0, nC1, ..... nCn are called binomial coefficients. nC0, nC2 nC4, ..... are called even binomial coefficients. nC1, nC3, nC5 .... are called odd binomial coefficients. |
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| Note 3: |
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| In case of no ambiguity, the binomial coefficients nC0, nC1, ..... nCn are written as C0, C1, ..... Cn. |
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| Note 4: |
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| The last term in L.H.S of (ii) and (iii)will depend on the fact as to whether n is even or odd. |
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| Working rules for solving problems: |
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Rule 1: If n  N, then |
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Rule 2: If n  N, then |
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Rule 3: If n  N, then |
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