Some Results on Binomial Coefficients and Negative or Fractional Indices

Some Results on Binomial Coefficients and Negative or Fractional Indices - Mathematics (Maths) Class 11 - Commerce

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 + C0C+ C0C3 + ... + C0C+ C1C2 + C1C3 + C1C+ 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)+ (C1 + C2)2 + (C1 + C3)2 + ....+ (C1 + Cn)2 + (C2 + C3)2 + (C+ 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.

The document Some Results on Binomial Coefficients and Negative or Fractional Indices | Mathematics (Maths) Class 11 - Commerce is a part of the Commerce Course Mathematics (Maths) Class 11.
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FAQs on Some Results on Binomial Coefficients and Negative or Fractional Indices - Mathematics (Maths) Class 11 - Commerce

 1. What are binomial coefficients and how are they calculated?
Ans. Binomial coefficients are the numbers that appear in the expansion of binomial expressions raised to a power. They represent the coefficients of the terms in the expansion. Binomial coefficients can be calculated using the formula: nCk = n! / (k!(n-k)!) Where n is the total number of items and k is the number of items chosen at a time.
 2. Can binomial coefficients have negative or fractional indices?
Ans. No, binomial coefficients are defined only for non-negative integer values of n and k. Negative or fractional indices are not applicable in the context of binomial coefficients.
 3. What are some important properties of binomial coefficients?
Ans. Some important properties of binomial coefficients include: - Symmetry: nCk = nC(n-k). This property reflects the symmetry in choosing k items out of n. - Pascal's Rule: (n+1)C(k+1) = nCk + nC(k+1). This rule allows for the calculation of binomial coefficients using smaller values. - Sum of Rows: The sum of the coefficients in each row of Pascal's Triangle is equal to 2^n, where n is the row number.
 4. How are binomial coefficients related to combinatorics and probability?
Ans. Binomial coefficients are closely related to combinatorics and probability. They are used to calculate the number of combinations or ways to choose a certain number of items out of a larger set. In probability, they are used in the calculation of binomial probabilities, which involve the probability of a specific number of successes in a fixed number of independent Bernoulli trials.
 5. Can binomial coefficients be used in real-world applications?
Ans. Yes, binomial coefficients have various real-world applications. They are used in fields such as statistics, genetics, physics, and computer science. For example, they can be used to calculate the probability of getting a certain number of heads in a series of coin flips, to model the genetic inheritance of traits, or to analyze the efficiency of algorithms.

Mathematics (Maths) Class 11

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Mathematics (Maths) Class 11

157 videos|210 docs|132 tests

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