**Binomial Expansion Practice Worksheet :**

Here we are going to see how to find expansion using binomial theorem.

**Binomial expansion for (x + a) ^{n} is,**

nc_{0}x^{n}a^{0 }+ nc_{1}x^{n-1}a^{1 }+ nc_{2}x^{n-2}a^{2 }+ .........+ nc_{n}x^{n-n}a^{0(}

If X is a set containing n elements, then we know that ^{n}C_{r} is the number of subsets of X having exactly r elements. So by adding ^{n}C_{r} for r = 0, 1, 2, . . . , n we get the number of subsets of X. So by using the above identity we see that a set of n elements has 2^{n} subsets.

(1) Expand (i) [2x^{2} − (3/x)]^{3 }Solution

(ii) (2x^{2} − 3√1 − x^{2})^{4 }+ (2x^{2} + 3√1 − x^{2})^{4 }Solution

(2) Compute (i) 102^{4} (ii) 99^{4 }(iii) 9^{7 }Solution

(3) Using binomial theorem, indicate which of the following two number is larger: (1.01)^{1000000}, 10000.

(4) Find the coefficient of x^{15} in (x^{2} + (1/x^{3}))^{10 }Solution

(5) Find the coefficient of x^{6} and the coefficient of x^{2} in (x^{2} - (1/x^{3}))^{6 }Solution

(6) Find the coefficient of x^{4} in the expansion of (1 + x^{3})^{50}(x^{2} + 1/x)^{5}. Solution

(7) Find the constant term of (2x^{3} - (1/3x^{2}))^{5 }Solution

(8) Find the last two digits of the number 3^{600 }Solution

(9) If n is a positive integer, show that, 9^{n+1} − 8n − 9 is always divisible by 64. Solution

(10) If n is an odd positive integer, prove that the coefficients of the middle terms in the expansion of (x + y)^{n} are equal. Solution

(11) If n is a positive integer and r is a non negative integer, prove that the coefficients of x^{r} and x^{n−r }in the expansion of (1 + x)^{n} are equal Solution

(12) If a and b are distinct integers, prove that a − b is a factor of a^{n} − b^{n}, whenever n is a positive integer. [Hint: write a^{n} = (a − b + b)^{n} and expand] Solution

(13) In the binomial expansion of (a + b)^{n}, the coefficients of the 4^{th} and 13^{th} terms are equal to each other, find n. Solution

(14) If the binomial coefficients of three consecutive terms in the expansion of (a + x)^{n} are in the ratio 1 : 7 : 42, then find n. Solution

(15) In the binomial coefficients of (1 + x)^{n}, the coefficients of the 5th, 6th and 7th terms are in AP. Find all values of n. Solution

(16)

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