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Class 11 Maths Chapter 8 Binomial Theorem
Binomial Theorem for Positive Integer
If n is any positive integer, then
This is called binomial theorem.
Here, nC0, nC1, nC2, … , nno are called binomial coefficients and
n
Cr = n! / r!(n – r)! for 0 ≤ r ≤ n.
Properties of Binomial Theorem for Positive Integer
(i) Total number of terms in the expansion of (x + a)n is (n + 1).
(ii) The sum of the indices of x and a in each term is n.
(iii) The above expansion is also true when x and a are complex numbers.
(iv) The coefficient of terms equidistant from the beginning and the end are equal. These
coefficients are known as the binomial coefficients and
n
Cr = nCn – r, r = 0,1,2,…,n.
(v) General term in the expansion of (x + c)n is given by
Tr + 1 = nCrxn – r ar.
(vi) The values of the binomial coefficients steadily increase to maximum and then steadily
decrease .
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(vii)
(viii)
(ix) The coefficient of xr in the expansion of (1+ x)n is nCr.
(x)
(xi) (a)
(b)
(xii) (a) If n is odd, then (x + a)n + (x – a)n and (x + a)n – (x – a)n both have the same number of
terms equal to (n +).
(b) If n is even, then (x + a)n + (x – a)n has (n +) terms. and (x + a)n – (x – a)n has (n / 2)
terms.
(xiii) In the binomial expansion of (x + a)n, the r th term from the end is (n – r + 2)th term
from the
beginning.
(xiv) If n is a positive integer, then number of terms in (x + y + z) n is (n + l)(n + 2) / 2.
Class 11 Maths Chapter 8 Binomial Theorem
Binomial Theorem for Positive Integer
If n is any positive integer, then
This is called binomial theorem.
Here, nC0, nC1, nC2, … , nno are called binomial coefficients and
n
Cr = n! / r!(n – r)! for 0 ≤ r ≤ n.
Properties of Binomial Theorem for Positive Integer
(i) Total number of terms in the expansion of (x + a)n is (n + 1).
(ii) The sum of the indices of x and a in each term is n.
(iii) The above expansion is also true when x and a are complex numbers.
(iv) The coefficient of terms equidistant from the beginning and the end are equal. These
coefficients are known as the binomial coefficients and
n
Cr = nCn – r, r = 0,1,2,…,n.
(v) General term in the expansion of (x + c)n is given by
Tr + 1 = nCrxn – r ar.
(vi) The values of the binomial coefficients steadily increase to maximum and then steadily
decrease .
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(vii)
(viii)
(ix) The coefficient of xr in the expansion of (1+ x)n is nCr.
(x)
(xi) (a)
(b)
(xii) (a) If n is odd, then (x + a)n + (x – a)n and (x + a)n – (x – a)n both have the same number of
terms equal to (n +).
(b) If n is even, then (x + a)n + (x – a)n has (n +) terms. and (x + a)n – (x – a)n has (n / 2)
terms.
(xiii) In the binomial expansion of (x + a)n, the r th term from the end is (n – r + 2)th term
from the
beginning.
(xiv) If n is a positive integer, then number of terms in (x + y + z) n is (n + l)(n + 2) / 2.
