Pascal’s triangle

Let Us Learn about Pascal's triangle.

The coefficients of the expansions are arranged in an array. This array is
called Pascal’s triangle.

One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher).

Patterns Within the Triangle

Binomial Coefficient

Let Us Understand What Is Binomial Coefficient.

the binomial coefficient is the coefficient of the x ^k term in the polynomial expansion of the binomial power (1 + x) ⁿ.

A binomial coefficient equals the number of combination of r items that can be selected from a set of n items. It also represents an entry in Pascal's triangle. These numbers are called binomial coefficients because they are coefficients in the binomial theorem.

Example:

Binomial Theorem

Let Us Learn What Is Binomial.


The sum of two monomial is called Binomial.

Binomial Theorem for Positive Integral Indices

Let us have a look at the following identities done earlier:

(a+ b)0 = 1 a + b ≠ 0
(a+ b)1 = a + b
(a+ b)2 = a2 + 2ab + b2
(a+ b)3 = a3 + 3a2b + 3ab2 + b3
(a+ b)4 = (a + b)3 (a + b) = a4 + 4a3b + 6a2b2 + 4ab3 + b4

In these expansions, we observe that

(i) The total number of terms in the expansion is one more than the index. For
example, in the expansion of (a + b)2 , number of terms is 3 whereas the index of
(a + b)2 is 2.

(ii) Powers of the first quantity ‘a’ go on decreasing by 1 whereas the powers of the
second quantity ‘b’ increase by 1, in the successive terms.

(iii) In each term of the expansion, the sum of the indices of a and b is the same and
is equal to the index of a + b.

Examples Using Binomial Theorem

Factorial Notation

Let Us Learn About Factorial Notation.


Question:What is Factorial Notation?
Answer: Let n be a positive integer. The continued product of first n natural numbers is called factorial n and is denoted as n

The notation n! represents the product of first n natural
numbers, i.e., the product 1 × 2 × 3 × . . . × (n – 1) × n is denoted as n!. We read this
symbol as ‘n factorial’. Thus, 1 × 2 × 3 × 4 . . . × (n – 1) × n = n !
1 = 1 !
1 × 2 = 2 !
1× 2 × 3 = 3 !
1 × 2 × 3 × 4 = 4 ! and so on.

We define 0 ! = 1
We can write 5 ! = 5 × 4 ! = 5 × 4 × 3 ! = 5 × 4 × 3 × 2 !
= 5 × 4 × 3 × 2 × 1!

Clearly, for a natural number n
n ! = n (n – 1) !
= n (n – 1) (n – 2) ! [provided (n ≥ 2)]
= n (n – 1) (n – 2) (n – 3) ! [provided (n ≥ 3)]
and so on.

Permutations when all the objects are distinct

Topic: Permutations are solved when all the objects are distinct

Question: How to Permutations are solved when all the objects are distinct?
Answer: The number of permutations of n different objects taken r at a time,
where 0 < r ≤ n and the objects do not repeat is n ( n – 1) ( n – 2). . .( n – r + 1), which is denoted by nPr. Proof There will be as many permutations as there are ways of filling in r vacant

places . . . by ← r vacant places →

the n objects. The first place can be filled in n ways; following which, the second place
can be filled in (n – 1) ways, following which the third place can be filled in (n – 2)
ways,..., the rth place can be filled in (n – (r – 1)) ways. Therefore, the number of
ways of filling in r vacant places in succession is n(n – 1) (n – 2) . . . (n – (r – 1)) or
n ( n – 1) (n – 2) ... (n – r + 1)

This expression for nPr is cumbersome and we need a notation which will help to
reduce the size of this expression. The symbol n! (read as factorial n or n factorial )
comes to our rescue. In the following text we will learn what actually n! means.