The Binomial Coefficients

Let’s keep it simple. What are binomial coefficients exactly? Well, they count certain ways of grouping items. The binomial coefficient C(n, k) counts the number of ways to form an unordered collection of k items chosen from a collection of n distinct items.

For example, if you wanted to make a 2-person committee from a group of four people say Ana (A),Bob(B),Charles(C) and Diny(D) the number of ways to do this is C(4, 2). Incidentally, there are 6 ways — just label the people A, B, C, D, and list all the two-letter sets: AB, AC, AD, BC, BD, and CD (order doesn’t matter, so BA is not any different than AB). So, we now know that C(4, 2) = 6.

There are many different ways that can be used to represent the binomial coefficients. All of these are equal to C(n, m) and may be read as: n choose m.

Briefly,one can say that they are a family of positive integers that occur as coefficients in the binomial theorem.

A binomial coefficient C(n, m) can be defined as the coefficient of x^m in the expansion of (1 + x)^n.

A binomial coefficient C(n, m) also gives the number of ways in which m objects can be chosen from among n objects more formally, the number of m-element subsets (or k-combinations) of an n-element set.

Blaise Pascal’s work

The Pascal’s Triangle

History

It is named after the 17th-century French mathematician Blaise Pascal, but it is far older. A Chinese mathematician ,Jia Xian, made a triangular representation for the coefficients back way earlier in the 11th century. His triangle was further studied and popularized by another Chinese mathematician Yang Hui , and after his name this triangle is also called as the Yang Hui triangle in China.

The triangle displays many interesting patterns. For example, drawing parallel diagonals and adding the numbers on each line together gives us the Fibonacci sequence.

Also if all the positions containing odd numbers are shaded black and all the positions containing even numbers are shaded white, a fractal known as the Sierpiński gasket is formed.

PASCALS’S TRIANGLE:

It’s a triangular arrangement of numbers in triangular shape that gives the coefficients in the expansion of a binomial expression, of the form (x + y)n.

The first row gives us the binomial coefficients of(x+y) raised to the power 1 in the first row, the second row gives the binomial coefficients of (x+y) to the power of 2 and similarly the nth row will give the binomial coefficients of (x+y) raised to the power n .

Disadvantages of Pascal’s Triangle

The Pascal’s Triangle is effective only when the expression is raised to a lower power because it’s very tedious and time consuming.

Therefore, we prefer using the formula for the same.

Working Formula for Binomial Coefficients

We can use the given formula for the same. For example, say we need to find (5

3) . So:-

— — — — — — C(7,3)

Properties

The total number of terms in the expansion of (x+y)^n is n+1.

Over here, nC0,nC1,nC2…..nCn are called the binomial coefficients of (x+y)^n . nC0,nC1,nC2……..nCn can also be represented by C0,C1,C2,C3 ………..Cn.

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The binomial coefficients which are equidistant from the beginning and from the ending are equal i.e. nC0 = nCn, nC1 = nCn-1 , nC2 = nCn-2 ,….. etc.

Useful Formulas:

• (x + y)n + (x−y)n = 2[C0 xn + C2 xn-1 y2 + C4 xn-4 y4 + …]
• (x + y)n — (x−y)n = 2[C1 xn-1 y + C3 xn-3 y3 + C5 xn-5 y5 + …]
• (1 + x)n = nΣr-0 nCr . xr = [C0 + C1 x + C2 x2 + … Cn xn]
• (1+x)n + (1 − x)n = 2[C0 + C2 x2+C4 x4 + …]
• (1+x)n − (1−x)n = 2[C1 x + C3 x3 + C5 x5 + …]

Properties

• C0 + C1 + C2 + … + Cn = 2n
• C0 + C2 + C4 + … = C1 + C3 + C5 + … = 2n-1
• C0 — C1 + C2 — C3 + … +(−1)n . nCn = 0
• nC1 + 2.nC2 + 3.nC3 + … + n.nCn = n.2n-1
• C1 − 2C2 + 3C3 − 4C4 + … +(−1)n-1 Cn = 0 for n > 1
• C02 + C12 + C22 + …Cn2 = [(2n)!/ (n!)2]

Application of Binomials

1 Combinatorics

Combinatorics as it name suggests is the study of possible combinations .It helps us to realize the number of ways in which we can choose anything from a given thing.

In combinatorics, the binomial coefficient represents the number of possible combinations of m objects from k.

Example: The possible ways to choose 3 wooden boxes from a carton with 10 wooden boxes can be represented by:

(10C3): 10.9.8/3.2.1

Example: There is a box of colorful balls containing one red ball, one blue ball, one orange ball and one green ball. The choice of two different balls from the box is a combination, and the number of possible choices are represented by

Example:- There are 10 children in your class but you can only give any 5 of them the chocolates .In how many different combinations can you distribute the chocolates among your friends could you invite?

The number of combinations to distribute the chocolate among 5 of your friends out of 10 is given by 10C5 = 252. We used combinations because in this situation it is order independent ,that is, it doesn’ matter in which order we decide to distribute the sweets .

2 Algebra

According to Binomial Theorem

Using the above formula , to find (x+y)³, we will proceed as follows:

3 Probability

Combinatorics , binomial coefficients , algebra and probability are closely related to each other. All of these are used in finding the chances of occurrence of an event.

P(X) = probability that X happens = number of outcomes where X happens /

total number of possible outcomes

Example :- A 5 digit PIN is selected. What is the probability that there are no repeated digits?

Solution :-The probability of no repeated digits are the number of 5 digit PINs with no repeated digits divided by the total number of 5 digit PINs. This probability is:

10P5/10⁵