SpongeBob SquarePants.

°•SpongeBob once said: What if i break your trust someday?
Patrick answered: Trusting you is my decision. Proving me wrong is your choice.°
                           
      

Introduction
Binomial Theorem is the method of expanding an expression that has been raised to a finite power. In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.

Consider the expansion of (x + y)5

(x + y)5 = (x + y)(x + y)(x + y)(x + y)(x + y)

The first term is obtained by multiplying the xs in the five brackets, there is only one way to doing this.

(x + y)n = xn + nxn-1y + n(n-1)/2! xn-2y2 + n(n-1)(n-2)/3! xn-3y3 + ...n(n-1)(n-2)...(n-r+1)/r! xn-ryr + ...yn

It can be shown that the binomial formula holds for positive, negative, integral or any rational value of n, provided there is a restriction on the values of x and y in the expansion of (x + y)n. We shall however consider only the binomial expansion formula for a positive integral n.

Elements of a Set

A set can be defined as a group or a collection of well-defined objects, numbers or alphabets e.g. collection of books, shoes, cooking utensils, alphabets, clothes. A set is denoted by capital letters such as A, B, C, etc. while small letters are used to denote the elements e.g a, b, c, etc.

Elements of a Set: These are the elements or members of a given set. The elements are separated by commas and enclosed by a curly bracket {}.

e.g P = {1,2,3,4,5}, 3 is an element of P.

Example: List the elements in each of the following sets

P = {Even numbers from 1 to 10}

B = {Odd numbers from 10 to 20}

Solution

P = {2, 4, 6, 8, 20}

B = {11, 13, 15, 17, 19}
Cardinality of a Set

This is the number of elements in a set. In order words, it is the measure of the number of elements in the cell.

Examples:

Given that A = {all the months of the year}, B = {all the months of the year that begins with letter J}

(i)List all the elements of A., (ii) List the members of B, (iii)What is n(A)?, (iv) What is n(A) x n(B)?

Solution

(i) A = {January, February, March, April, May, June, July, August, September, October, November, December}

(ii) B = {January, June, July}, (iii) n(A) = 12, (iv) n(A) x n(B) = 12 x 3 = 36

Set Notation

Sets can be denoted algebraically using inequalities and other symbols. E.g. A = {x: -6 ≤ x ≤ 2, x is an integer}

Example: Highlight the members of the following sets

There was then no king in Edom: a deputy was king. Jehoshaphat made ships of Tharshish to go to Ophir for gold: but they went not; for the ships were broken at Eziongeber.

Then king Asa made a proclamation throughout all Judah; none was exempted: and they took away the stones of Ramah, and the timber thereof, wherewith Baasha had builded; and king Asa built with them Geba of Benjamin, and Mizpah.

Then king Asa made a proclamation throughout all Judah; none was exempted: and they took away the stones of Ramah, and the timber thereof, wherewith Baasha had builded; and king Asa built with them Geba of Benjamin, and Mizpah.

Then king Asa made a proclamation throughout all Judah; none was exempted: and they took away the stones of Ramah, and the timber thereof, wherewith Baasha had builded; and king Asa built with them Geba of Benjamin, and Mizpah.

Then king Asa made a proclamation throughout all Judah; none was exempted: and they took away the stones of Ramah, and the timber thereof, wherewith Baasha had builded; and king Asa built with them Geba of Benjamin, and Mizpah.

And king Solomon made a navy of ships in Eziongeber, which is beside Eloth, on the shore of the Red sea, in the land of Edom.