D. Some Results on Binomial Coefficients
(a) C0 + C1 + C2 + ............+ Cn = 2n
(b) C0 + C2 + C4 + ............= C1 + C3 + C5 + .......... = 2n -1
Remember : (2n) ! = 2n. n! [1.3.5........(2n -1)]
Ex.8 If (1 + x)n = C0 + C1x + C2x2 +.................+ Cnxn then show that the sum of the products of the taken two at a time represents by :
Sol.
Since (C0 + C1 + C2 + ........+Cn – 1 + Cn)2 = C02 + C12 + C22 + ...... + Cn-1 2+ Cn2 + .......+
2(C0C1 + C0C2 + C0C3 + ... + C0Cn + C1C2 + C1C3 + C1Cn + C2C3 + C2C4 + ... + C2Cn + ...... + Cn–1Cn )
Ex.9 If (1 + x)n = C0 + C1x + C2x2 + .........+Cnxn then prove that
Sol.
(C0 + C1)2 + (C0 + C2)2 + ....+ (C0 + Cn)2 + (C1 + C2)2 + (C1 + C3)2 + ....+ (C1 + Cn)2 + (C2 + C3)2 + (C2 + C4)2 + ........+ (C2 + Cn)2 + .........+ (Cn–1 + Cn)2
E. Binomial theorem for negative or fractional indices
If n ∈ Q, then (1 + x)n = provided |x| < 1.
Note :
(i) When the index n is a positive integer the number of terms in the expansion of (1 + x)n is finite i.e. (n + 1) & the coefficient of successive terms are : nC0, nC1, nC2, .........., nCn
(ii) When the index is other than a positive integer such as negative integer or fraction, the number of terms in the expansion of (1 + x)n is infinite and the symbol nCr cannot be used to denote the coefficient of the general term.
(iii) Following expansion should be remembered (|x| < 1)
(a) (1 + x)–1 = 1 – x + x2 – x3 + x4 – .........∝
(b) (1 – x)–1 = 1 + x + x2 + x3 + x4 + .........∝
(c) (1 + x)–2 = 1 –2x + 3x2 – 4x3 + .........∝
(d) (1 – x)–2 = 1 + 2x + 3x2 + 4x3 + .........∝
(iv) The expansions in ascending powers of x are only valid if x is 'small'. If x is large i.e. |x| > 1 then we may find it convenient to expand in powers of 1/x, which then will be small.
1. What are binomial coefficients and how are they calculated? | ![]() |
2. Can binomial coefficients have negative or fractional indices? | ![]() |
3. What are some important properties of binomial coefficients? | ![]() |
4. How are binomial coefficients related to combinatorics and probability? | ![]() |
5. Can binomial coefficients be used in real-world applications? | ![]() |
157 videos|210 docs|132 tests
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157 videos|210 docs|132 tests
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