**Binomial Expansion Practice Questions :**

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

**Question 1 :**

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 :**

a^{n} = [(a - b) + b]^{n}

a^{n} = ^{n}C_{0}(a-b)^{n}+^{n}C_{1}(a-b)^{n-1}b^{1}+^{n}C_{2}(a-b)^{n-2}b^{2}+^{n}C_{n-1}(a-b)b^{n-1}+ ^{n}C_{n}b^{n}

a^{n} - b^{n} = (a-b)^{n}+^{n}C_{1}(a-b)^{n-1}b^{1}+^{n}C_{2}(a-b)^{n-2}b^{2}+^{n}C_{n-1}(a-b)b^{n-1}

a^{n} - b^{n}

= (a-b) { (a-b)^{n-1 }+ ^{n}C_{1}(a-b)^{n-2}b + ^{n}C_{2}(a-b)^{n-3}b^{2 }+ ^{n}C_{n-1}b^{n-1 }}

Hence a − b is a factor of a^{n} − b^{n}, whenever n is a positive integer.

**Question 2 :**

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 :**

The coefficients of the fourth ad thirteenth terms in the binomial expansion of (a + b)^{n} are ^{n}C_{3} and ^{n}C_{12} respectively.

Coefficient of 4^{th} term in (a+b)^{n} = coefficient of 13^{th} term in (a + b)^{n}

^{n}C_{3} = ^{n}C_{12}

If ^{n}C_{x} = ^{n}Cy ==> x = y or x + y = n

n = 15

**Question 3 :**

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 :**

Let the three consecutive terms be r^{th}, (r+1)^{th} and (r+2)^{th} terms.Their coefficients in the expansion of (1+x)n are ^{n}C_{r-1}, ^{n}C_{r} and ^{n}C_{r+1} respectively. It is given that,

^{n}C_{r-1 }: ^{n}C_{r : }^{n}C_{r+1 }= 1 : 7 : 42

^{n}C_{r-1 /} ^{n}C_{r } = 1 / 7

r/(n-r+1) = 1/7

n - 8r + 1 = 0 ----(1)

^{n}C_{r : }^{n}C_{r+1 }= 7/42

r+1/n-r = 1/6

n - 7r - 6 = 0 -----(2)

(1) - (2) ==> (n - 8r + 1) - (n - 7r - 6) = 0

n - n - 8r + 7r + 1 + 6 = 0

-r + 7 = 0 ==> r = 7

By applying the value of r in the 1st equation, we get

n - 8(7) + 1 = 0

n - 56 + 1 = 0

n = 55

Hence the values of n and r are 55 and 7 respectively.

**Question 4 :**

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 :**

The coefficients of 5^{th}, 6^{th} and 7^{th} terms in the binomial expansion of (1+x)^{n} are ^{n}C_{4}, ^{n}C_{5} and ^{n}C_{6} respectively,We are given that

2 ^{n}C_{5 } = ^{n}C_{4 + }^{n}C_{6}

2 = (^{n}C_{4})/(^{n}C_{5}) + (^{n}C_{6})/(^{n}C_{5})

2 = 5/(n-4) + (n-5)/6

2 = [30 + (n-4)(n-5)]/[6(n-4)]

12(n-4) = = [30 + (n^{2} - 4n - 5n + 20)]

12n - 48 = 30 + n^{2} - 9n + 20

n^{2} - 9n + 20 - 12n + 48 + 30 = 0

n^{2} - 21n + 98 = 0

(n - 14) (n - 7) = 0

n - 14 = 0 n - 7 = 0

n = 14, n = 7

After having gone through the stuff given above, we hope that the students would have understood "Binomial Expansion Practice Questions".

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