What is 4th term of the Binomial expansion of (10 + 10)10 ?
(a+x)^10 maybe ?
= a^10 + 10a^9 x^1 + 10*9/2 (a^8 x^2 )
+ [10(9)(8) /(3*2)] a^7 x^3 +..............
which fourth term is
120 a^7 x^3
To find the 4th term of the binomial expansion of (10 + 10)^10, we can use the binomial theorem.
The binomial theorem states that for any positive integer n and any real numbers a and b:
(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n
where C(n, k) represents the combination of choosing k items from a set of n items.
In our case, a = 10, b = 10, and n = 10. We want to find the term with k = 4.
Using the formula for combination, C(n, k) = n! / (k! * (n-k)!), we can calculate the coefficients:
C(10, 4) = 10! / (4! * (10-4)!) = 10! / (4! * 6!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210.
Now, we can substitute these values into the formula:
(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n
(10 + 10)^10 = C(10, 0) * 10^10 * 10^0 + C(10, 1) * 10^9 * 10^1 + C(10, 2) * 10^8 * 10^2 + C(10, 3) * 10^7 * 10^3 + C(10, 4) * 10^6 * 10^4 + ... + C(10, 9) * 10^1 * 10^9 + C(10, 10) * 10^0 * 10^10
Simplifying this expression, we find that the 4th term is:
C(10, 4) * 10^6 * 10^4 = 210 * 10^6 * 10^4 = 210 * 10^10 = 2,100,000,000,000.
To find the 4th term of the binomial expansion of (10 + 10)^10, we can use the formula for the term of the binomial expansion:
Term(k) = (nCk) * (a)^(n-k) * (b)^k
Here, n is the power of the binomial, k is the term number (starting from 0), a and b are the coefficients of the binomial. In this case, a = 10 and b = 10.
In the given expression, (10 + 10)^10, we have n = 10, a = 10, and b = 10. We need to find the 4th term, which corresponds to k = 3.
Plugging these values into the formula, we get:
Term(3) = (10C3) * (10)^(10-3) * (10)^3
Now, let's calculate the individual components:
(10C3) = 10! / (3! * (10-3)!) = 10! / (3! * 7!)
Expand the factorials:
(10C3) = (10 * 9 * 8 * 7!)/(3 * 2 * 1 * 7!) = (10 * 9 * 8) / (3 * 2 * 1)
Calculating the multiplication:
(10C3) = 120
Now, let's calculate the remaining part:
(10)^(10-3) = 10^7
(10)^3 = 1000
Now, let's put everything together and calculate the 4th term:
Term(3) = (10C3) * (10)^(10-3) * (10)^3
= 120 * 10^7 * 1000
Calculating this expression:
Term(3) = 120 * 10^7 * 1000 = 1,200,000,000
Therefore, the 4th term of the binomial expansion of (10 + 10)^10 is 1,200,000,000.