Hello khushal , as you want here we provides you solution of your query i.e.
“applications for Binomial Theorem” as

In elementary algebra, the binomial theorem (or binomial expansion)
describes the algebraic expansion of powers of a binomial. According to the
theorem, it is possible to expand the power (x + y)n into a sum involving
terms of the form a xb yc, where the exponents b and c are nonnegative
integers with b + c = n, and the coefficient a of each term is a specific
positive integer depending on n and b

Expansion of Binomials

The binomial theorem can be used to find a complete expansion of a power
of a binomial or a particular term in the expansion. Here are examples of
each.

Example: Expand (1 + x)4.

Solution: A straightforward application of the binomial theorem gives us:

(1 + x)4 = C(4, 0)14 + C(4, 1)13x + C(4, 2)12x2 + C(4, 3)1x3 + C(4,
4)x4

= 1 + 4x + 6x2 + 4x3 + x4

Example: Find the 7th summand of (1 − x)10.

Solution: Now here is one we definitely do not want to expand completely
in order to find the 7th summand! So what do we do? Let's think about the
properties of the summands of the binomial expansion of (a + b)n (click here
for a reminder):

The first summand is C(n, 0)anb0 = an.

The second summand is C(n, 1)an − 1b1 = nan − 1b.

In general, the kth summand will contain C(n, k − 1) an − (k − 1)bk − 1.
Notice the coefficient is not C(n, k) but C(n, k − 1). This is analogous to the
fact that we call the first number in each row of Pascal's Triangle entry 0,
not entry 1.

So the 7th of (a + b)10 will be: C(10, 6) a10 − (7 − 1)b7 − 1 =
(10!/(6!4!))a4b6. = 210a4b6.

In our case, a = 1 and b = −x, so the summand we are looking for will be:

210 14(−x)6 = 210(−1)6x6 = 210x6.

Much better than carrying out the full expansion!

In this exercise you are to use binomial coefficients to find a particular
coefficient in a binomial expansion. Each answer should be entered in integer
form. Decide first which summand is called for. For example x2 appears in
the second summand of (x + 2)3 but in the third of (4 − x)3. Click "New" for
a new problem. Type your answer in the typing area.