**Example 1:** Expand (5x – 4)^{10}

**Sol: **(5x – 4)^{10} = ^{10}C0 (5x)^{10–0}(–4)^{0} + ^{10}C1 (5x)^{10–1}(–4)^{1}

+ ^{10}C_{2} (5x)^{10–2}(–4)^{2} + ^{10}C_{3} (5x)^{10–3}(–4)^{3}

+ ^{10}C_{4} (5x)^{10–4}(–4)^{4} + ^{10}C_{5} (5x)^{10–5}(–4)^{5}

+ ^{10}C_{6} (5x)^{10–6}(–4)^{6} + ^{10}C_{7} (5x)^{10–7}(–4)^{7}

+ ^{10}C_{8} (5x)^{10–8}(–4)^{8} + ^{10}C_{9} (5x)^{10–9}(–4)^{9}

+ ^{10}C_{10} (5x)^{10–10}(–4)^{10}

**Example 2:**Find the third term in the expansion of (3 + y)^{6}

**Sol:** As expansion is of the form (a + x)^{n}, so r^{th} term

= a^{n–r+1} x^{r–1} [{n(n–1) (n – 2) ... (n – r + 2)} ÷ (r – 1)!].

Here r = 3 and n = 6.

So 3^{rd} term of (3 + y)^{6} = 3^{6 – 3 + 1} . y^{3 – 1 } . [(6x5)/2]

=3^{4} . y^{2} . 15 = 1215 y^{2}

**Example 3:**Find the co-efficient of z4 in the expansion of (5 + z)8.

**Sol: **As expansion is of the form (a + x)^{n}, so rth term

= a^{n – r + 1} x^{r – 1} [{n (n – 1) (n – 2) ... (n – r + 2)} ÷ (r – 1)!].

z4 will come in 5^{th} term.

Hence we have to find the 5^{th} term of the expansion.

Here r = 5 and n = 8.

So 5^{th} term of (5 + z)^{8} =5 ^{8 – 5 + 1}. z^{5 – 1 }.

= 5^{4} . z^{4} . 70 = 625x70x4 = 43750 z^{4}

Hence coefficient is z^{4} is 43750.

**Example 4:** Find the co-efficient of p5 in the expansion of (p + 2)^{6}.

**Sol:** As expansion is of the form (x + a)^{n}, so r^{th} term

= x ^{n – r + 1} a ^{r – 1} [{n(n–1) (n – 2) ... (n – r + 2)} ÷ (r – 1)!].

So x^{5} will come when r = 2 and n = 6.

Hence we have to find the 2nd term of the expansion.

So r = 2 and n = 6.

So 2^{nd} term of (p + 2)^{6} = p^{6 – 2 + 1} . 2^{6 – 1 }.

= p^{5} . 2^{5} . 6 = 192 p^{5}

Hence coefficient of p^{5} is 192.

**Example 5:** For what value of x, the fifth term of the following expansion is equal to 105?

**Sol: ** (-1)^{6} . ^{10}c_{4} (1/2√x)^{6}(1/2)^{4} = 105 => x^{3} = ^{10}c_{4}/105.2^{10}

x = 1/8