**Algebra > Binomial Theorem Examples **

**13. Let us understand the Binomial Theorem concepts discussed above with the following numerous solved examples on each of the concepts and formulas:**

**1. Find the number of terms in the following binomial expansions:**

1. (3x + 4y )^{10} 2. (x – ^{1}/_{x} )^{17}

Solution:

We know the number of terms in a binomial expansion is always one more than one, i.e.,n + 1. In question 1, number of terms is 10 + 1, i.e., 11 and in question 2, the number of terms is 17 + 1, i.e., 18.

**2. Use the binomial theorem to expand the following binomial expansion or write all the terms in the following binomial expansion:**

1. (2x – 1/_{x})^{5}

*Solution:*

Let us use the binomial theorem to write all the terms in the two examples in this question.
Remember the binomial theorem? It is:

(x+y)^{n} = ^{n}c_{0}. x^{n} +^{n}c_{1}. x^{n-1}. y + ^{n}c_{2} x^{n-2}. y^{2} + ^{n}c_{2} . x^{n-3}. y^{3} + ……. + ^{n}cr. x^{n-r}. y^{r} + ……. + ^{n}c_{n} .y^{n}

Compare the given binomial expansion (2x – ^{1}/_{x})^{5} with the standard form (x+y)^{n}.

So we see that we must write in the binomial theorem:

2x for x and – (^{1}/_{x} ) for y.

(2x – ^{1}/_{x})^{5} = ^{5}c_{0} . (2x)^{5} + ^{5}c_{1} . (2x)^{4}. y + ^{5}c_{2} . (2x)^{3}. (y)^{2} + ^{5}c_{3} .(2x)^{2} (y)^{3} +
^{5}c_{4} .(2x ) (y)^{4} + ^{5}c_{5} .(y)^{5}

**Write the following values of the binomial coefficients above**

^{5}c_{0} = 1, ^{5}c_{1} = 5, ^{5}c_{2} = 10,^{5}c_{3} = 10, ^{5}c_{4} = 5, ^{5}c_{5} = 1.

= 1. (2)^{5}. (x)^{5} + 5.(2)^{4}.(x)^{4}.(y) + 10.(2)^{3}.(x)^{3}.(y)^{2} + 10. (2)^{2}.(x)^{2}.(y)^{3} + 5.(2x).(y)^{4} + 1.(y)^{5}

= 32.x^{5} + 80.x^{4}y + 80 . x^{3} y^{2} + 40.x^{2}y^{3} + 10xy^{4} + y^{5}

**3. Write the tenth term in the binomial expansion (4x ^{3} – ^{1}/_{x²} )^{13}**

*solution:*

T_{r + 1} = ^{n}c_{r} .x^{ n – r} . y^{ r}

We need to find the 10th term in the given expansion i.e., T_{10} = T_{9 + 1}

So r is 9. ( also, recall that the value of r is always one less than the term)

Again, compare the standard form of the binomial expansion with the given form to write x and y in the respective places

On comparing, we see that x = 4x^{3} and y = ^{– 1}/_{x²} and n = 13

So, the tenth term is:

T_{9 + 1} = T_{10} = ^{13}c_{9} . (4x^{3})^{13 – 9}.
( ^{– 1}/_{x²} )^{9}

= ^{13}c_{9} .(4)^{4} (x^{3})^{4} (-1)^{9} ( ^{1}/_{x²} )^{9}

{ (-1)^{9} = -1 and ^{13}c_{9} = ^{13!}/ _{9! 4!} = ^{( 13. 12. 11. 10. 9!)} / _{(9! . 24)} = 715}

= - 715 (4)^{4}. (x)^{12}.(x)^{-18} { (x)12.(x)-18 = x -6 }

= -183040. (x)^{-6}