D. Some Results on Binomial Coefficients
(a) C_{0} + C_{1} + C_{2} + ............+ C_{n} = 2^{n}
(b) C_{0} + C_{2} + C_{4} + ............= C_{1} + C_{3} + C_{5} + .......... = 2^{n 1}
Remember : (2n) ! = 2^{n}. n! [1.3.5........(2n 1)]
Ex.8 If (1 + x)^{n} = C_{0} + C_{1}x + C_{2}x^{2} +.................+ C_{n}x^{n} then show that the sum of the products of the taken two at a time represents by :
Sol.
Since (C_{0} + C_{1} + C_{2} + ........+C_{n – 1} + C_{n})^{2} = C_{0}^{2} + C_{1}^{2} + C_{2}^{2} + ...... + C_{n1} ^{2}+ C_{n}^{2} + .......+
2(C_{0}C_{1} + C_{0}C_{2 }+ C_{0}C_{3} + ... + C_{0}C_{n }+ C_{1}C_{2} + C_{1}C_{3} + C_{1}C_{n }+ C_{2}C_{3} + C_{2}C_{4} + ... + C_{2}C_{n} + ...... + C_{n–1}C_{n} )
Ex.9 If (1 + x)^{n} = C_{0} + C_{1}x + C_{2}x_{2} + .........+C_{n}x_{n} then prove that
Sol.
(C_{0} + C_{1})2 + (C_{0} + C_{2})2 + ....+ (C_{0} + C_{n})^{2 }+ (C_{1} + C_{2})^{2} + (C_{1} + C_{3})^{2} + ....+ (C_{1} + C_{n})^{2} + (C_{2} + C_{3})^{2} + (C_{2 }+ C_{4})^{2} + ........+ (C_{2} + C_{n})^{2} + .........+ (C_{n–1} + C_{n})^{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 : ^{n}C_{0}, ^{n}C_{1}, ^{n}C_{2}, .........., ^{n}C_{n}
(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 ^{n}C_{r} 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 + x^{2} – x^{3} + x^{4} – .........∝
(b) (1 – x)^{–1} = 1 + x + x^{2} + x^{3} + x^{4} + .........∝
(c) (1 + x)^{–2} = 1 –2x + 3x^{2} – 4x^{3} + .........∝
(d) (1 – x)^{–2} = 1 + 2x + 3x^{2} + 4x^{3} + .........∝
(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 realworld applications? 
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157 videos210 docs132 tests
