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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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