1. If n is a positive integer then n)ax( + = +++++ −−− rrnr

n22n2

n1n1

nn0

n axC...axCaxCxC nn

n aC...... + .

2. The expansion of n)ax( + contains (n + 1) terms.

3. In the expansion, the sum of the powers of x and a in each term is equal to n. 4. In the expansion, the coefficients n

n2

n1

n0

n C,,...C,C,C are called binomial coefficients and these are simply denoted by n210 C,,...C,C,C .

5. In the expansion, (r + 1)th term is called the general term. It is denoted by 1rT + . Thus 1rT + = rrn

rn axC − .

6. n)ax( + = ∑=

−n

0r

rrnr

n axC .

7. n)ax( − = ∑∑=

−

=

− −=−n

0r

rrnr

nnn

0r

rrnr

n axC)1()a(xC = nn

nn22n2

n1n1

nn0

n aC)1(...axCaxCxC −+−+− −−

8. =+ n)x1( =∑=

n

0r

rr

n xC =++++ nn

n22

n1

n0

n xC...xCxCC nn

2210 xC...xCxCC ++++

9. Middle term(s) in the expansion of (x + a)n :

i) If n is even then ⎟⎠⎞

⎜⎝⎛ + 1

2n th term is the middle term.

ii) If n is odd then 2

1n + th and 2

3n + th terms are the middle terms.

10. Numerically greatest term in the expansion of (1 + x)n :

i) If 1|x|

|x|)1n(+

+ = p, a positive integer then pth and (p+1)th terms are the numerically greatest terms

in the expansion of (1 + x)n.

ii) If 1|x|

|x|)1n(+

+ = p+F where p is a positive integer and 0 < F < 1 then (p+1)th term is the

numerically greatest term in the expansion of (1+x)n. 11. If Cr denotes r

n C then

i) nn210 2C...CCC =++++

ii) 1n531420 2...CCC...CCC −=+++=+++

12. i) nn

0rr

n 2C =∑=

ii) 0C)1(n

0rr

nr =−∑=

13. a⋅C0 + (a + d)⋅C1 + (a + 2d)⋅C2 +……+ (a + nd)⋅Cn = (2a + nd) ⋅2n–1.

14. 112321 )1(...32 −− +=⋅++⋅+⋅+ nn

n xnxCnxCxCC

Binomial Theorem

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Binomial Theorem

2

15. i) 1nn321 2nCn...C3C2C −⋅=⋅++⋅+⋅+

ii) 0......C3C2C 321 =−⋅+⋅−

16. i) 1nn

0rr

n 2.nC.r −

=

=∑ ii) 0C.r)1(n

0rr

nr =−∑=

17. xn

xxnC

xCxCCn

nn

)1(1)1(

1....

32

1221

0 +−+

=+

+++++

.

18. rnn2

nrn2r21r1r0 CCC.....CCCCCC −−++ =++++ = )!rn()!rn(

)!n2(+−

.

19. 22n

22

21

20 )!n(

)!n2(C...CCC =++++ .

20. Binomial Theorem for rational index : If n is a rational number and |x| < 1 then

1+nx + n32 )x1(....x!3

)2n)(1n(nx!2

)1n(n+=+

−−+

+ .

21. If |x| < 1 then i) (1 + x)–1 = ...x)1(...xxx1 rr32 +−++−+−

ii) (1 – x)–1 = ...x...xxx1 r32 ++++++ iii) (1+x)–2= ...x)1r()1(...x4x3x21 rr32 ++−++−+−

iv) (1 – x)–2 = ...x)1r(...x4x3x21 r32 +++++++

v) ...x!3

)2n)(1n(nx!2

)1n(nnx1)x1( 32n +−−

−−

+−=+ −

vi) ...x!3

)2n)(1n(nx!2

)1n(nnx1)x1( 32n +++

++

++=− −

vii) +⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛+=−

− 2qp

px

!2)qp(p

pxp1)x1( ...

px

!3)q2p)(qp(p

3

+⎟⎟⎠

⎞⎜⎜⎝

⎛++

22. If |x| < 1 and n is a positive integer, then i) ...xCxCxC1)x1( 3

3)2n(2

2)1n(

1nn ++++=− ++−

ii) ...xCxCxC1)x1( 33

)2n(22

)1n(1

nn +−+−=+ ++−

23. Coefficient of mx in the expansion of n

qp

xbax ⎟⎠⎞

⎜⎝⎛ + is the coefficient of 1rT + where r =

qpmnp

+− .

24. The term independent of x in the expansion of n

qp

xbax ⎟⎠⎞

⎜⎝⎛ + is 1rT + where r =

qpnp+

.

25. In the expansion xa

r1rn

TT,)ax(

r

1rn +−=+ + .

26. The general term in the expansion of np21 )x...xx( +++ is p21 n

nn2

n1

p21x...xx

!n!...n!n!n where

.nn...nn p21 =+++

Binomial Theorem

3

27. i) The coefficient of 1nx − in the expansion of (x – 1)(x – 2) … (x – n) is –n(n+1)/2. ii) The coefficient of 1nx − in the expansion of (x + 1)(x + 2) … (x + n) is n(n+1)/2. 28. i) If the coefficients of pth, qth terms in the expansion of (1 + x)n are equal then p+q = n+2. ii) If the coefficients of 1rr x,x + in the expansion of (a + x/b)n are equal then n = (r+1)(ab+1) –1.

29. i) If the coefficients of rth, (r+1)th, (r+2)th terms of (1 + x)n are in A.P. then n2 – (4r+1)n + 4r2=2. ii) If the coefficients of 1rr1r x,x,x +− are in A.P. then (n – 2r)2 = n + 2.

30. If n)ba( + = I + F were I, n are positive integers, 0 < F < 1, a2 – b = 1, then

i) I is an odd positive integer ii) (I + F)( I – F) = 1. 31. i) The number of terms in the expansion of nn )ax()ax( −++ is (n+2)/2 if n is even, (n+1)/2 if n is

odd. ii) The number of terms in the expansion of nn )ax()ax( −−+ is n/2 if n is even, (n+1)/2 if n

is odd. 32. Sum of the coefficients of (ax + by)n is (a + b)n. 33. If f(x) = (a0 + a1x + a2x2 +…+ amxm)n then i) Sum of the coefficients = f(1).

ii) Sum of the coefficients of even powers of x is 2

)1(f)1(f −+ .

iii) Sum of the coefficients of odd powers of x is 2

)1(f)1(f −− .

34. 2n

22

21

20 C).nda(...C).d2a(C).da(C.a +++++++ = .C).nda2(

21

nn2+

35. rnn2

nrn2r21r1r0 CCC...CCCCCC −−++ =++++ .

36. =++++ −− 0n

rm

2rn

2m

1rn

1m

rn

0m CC...CCCCCC r

)nm( C+ .

37. If n is a positive integer, then n)x1( −− = ...xCxCxC1 33

)2n(22

)1n(1

n ++++ ++

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