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The number of terms in the expansion of (2x  3y)^{8} is
Since this binomial is to the power 8, there will be nine terms in the expansion.
n = 10
Middle term = (n/2) + 1
= (10/2) + 1
= 6th term
T(6) = T(5+1)
= ^{10}C_{5}[(2x^{2})/3]^{5} [(3/2x^{2})]^{5}
= ^{10}C_{5}
= 252
In the expansion of (a+b)^{n}, N the number of terms is:
The total number of terms in the binomial expansion of (a + b)^{n} is n + 1, i.e. one more than the exponent n.
Find the value of r, if the coefficients of (2r + 4)^{th} and (r – 2)^{th} terms in the expansion of (1 + x)^{18} are equal.
If the coefficients of 7^{th} and 13^{th} terms in the expansion of (1 + x)^{n} are equal, then n is equal to
What is the coefficient of x^{5} in the expansion of (1x)^{6 }?
(1x)^{6}
=> (1x)^{(6/1)}
It is in the form of (1x)^{(p/q)}, p =6, q=1
(1x)^{(p/q)} = 1+p/1!(x/q)^{1} + p(p+q)/2!(x/q)^{2} + p(p+q)(p+2q)/3!(x/q)^{3} + p(p+q)(p+2q)(p+3q)/4!(x/q)^{4}........
= 1+6/1!(x/1)^{1} + 6(7)/2!(x/1)^{2} + 6(7)(8)/3!(x/1)^{3} + 6(7)(8)(9)/4!(x/1)^{4} +.......................
So, coefficient of x^{5} is (6*7*8*9*10)/120
= 252
In the expansion of the binomial expansion (a + b)^{n}, which of the following is incorrect ?
Correct Answer: d
Explanation: The coefficient of terms (x+a)^{n} equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients.
^{n}C_{r} = ^{n}C_{n – r}, r = 0,1,2,…,n.
The middle term in the expansion of (x + y)^{10} is the
Number of terms(n) = 10
Middle term = (n/2) + 1
= (10/2) + 1
= 5 + 1
= 6th term
If in the expansion of (1+x)^{20}, the coefficients of r^{th} and (r+4)^{th} terms are equal, then the value of r is equal to:
Coefficients of the rth and (r+4)th terms in the given expansion are C_{r−120} and ^{20}C_{r+3}.
Here,C_{r−120} = ^{20}C_{r+3}
⇒ r−1+r+3 = 20
[∵ if ^{n}C_{x }= ^{n}C_{y} ⇒ x = y or x+y = n]
⇒ r = 2 or 2r = 18
⇒ r = 9
The number of terms in the expansion of (x – y + 2z)^{7} are:
Here the number of terms can be calculated by:
= ((n+ 1) * (n+2)) /2
where, n =7
∴ Number of terms = 36
The number of terms in the expansion of (a + b + c)^{n} are:
No. of terms is ^{n+2}C_{2}
The general term in the expansion of (a  b)^{n} is
If a and b are real numbers and n is a positive integer, then:
(a  b)^{n} = ^{n}C_{0} a^{n} + ^{n}C_{1} a^{(n – 1)} b^{1} + ^{n}C_{2} a^{(n – 2)} b^{2}+ ...... + ^{n}C_{r} a^{(n – r)} b^{r+ ... +} ^{n}C_{n}b^{n},
The general term or (r + 1)th term in the expansion is given by:
T_{r + 1 }= (1)C_{r} a^{(n–r)} b^{r}
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