 Hello and welcome to the session. In this session first we will discuss binomial theorem. Expansion of a binomial for any positive integer n is given by a plus b whole to the power n is equal to nc0 multiplied by a to the power n plus nc1 multiplied by a to the power n minus 1 multiplied by b plus nc2 multiplied by a to the power n minus 2 multiplied by b to the power 2 plus and so on up to nc n minus 1 into a into b to the power n minus 1 plus ncn multiplied by b to the power n. Next we will discuss Pascal's Triangle, the coefficients of the expansions are arranged in an array. This array is called Pascal's Triangle. The expansion of binomials by Pascal's Triangle is a slightly lengthy process. We know that the notation submission k goes from 0 to n nck multiplied by a to the power n minus k into b to the power k stands for nc0 into a to the power n into b to the power 0 plus nc1 into a to the power n minus 1 into b to the power 1 plus and so on up to ncr into a to the power n minus r into b to the power r plus and so on up to ncn into a to the power n minus n into b to the power n. Where we have b to the power 0 is equal to 1 that is equal to a to the power n minus n and thus the theorem that is the binomial theorem can be stated as a plus b whole to the power n is equal to summation k goes from 0 to n n c k into a to the power n minus k into b to the power k. Then we have the coefficients n c r occurring in the binomial theorem are called the binomial coefficients and in the expansion of a plus b whole to the power n there are n plus 1 terms that is 1 more than the index as you can observe in the expansion of a plus b whole to the power n in the successive terms of the expansion the index of a goes on decreasing by unity that is the index of a in the first term is n then in the second term it's n minus 1 so it is decreasing by unity and in the last term we have 0 as the index of a and at the same time we have the index of b goes on increasing by unity in the first term the index of b is 0 then in the second term it's 1 in the third term it's 2 and in the last term it's n also we have in the expansion of a plus we whole to the power n sum of the indices of a and b is n in every term of the expansion like in the first term as you can see a has index n and b has index 0 so n plus 0 is n then again in the second term a has index n minus 1 and b has index 1 n minus 1 plus 1 is n so every term has the sum of indices of a and b as n now we shall consider some special cases like in the expansion of a plus we whole to the power n if we take a as x and b as minus y then we get x minus y whole to the power n is equal to n c 0 into x to the power n minus n c 1 into x to the power n minus 1 into y plus n c 2 into x to the power n minus 2 into y to the power 2 plus and so on up to minus 1 to the power n into n c n into y to the power n and when we take a as 1 and b as x we get 1 plus x whole to the power n is equal to n c 0 plus n c 1 into x plus n c 2 into x square plus and so on up to n c n into x to the power n the next is when we take a as 1 and b as minus x so we get 1 minus x whole to the power n is equal to n c 0 minus n c 1 into x plus n c 2 into x square minus and so on plus minus 1 whole to the power n into n c n into x to the power n let's expand the expression 1 minus 2x whole to the power 5 this expression is of the kind when we take a as 1 and b as minus x so in this expansion we will take x as 2x and n as 5 so we have 1 minus 2x whole to the power 5 is equal to 5 c 0 minus 5 c 1 into 2x plus 5 c 2 into 2x to the power 2 minus 5 c 3 into 2x to the power 3 plus 5 c 4 into 2x to the power 4 minus 5 c 5 into 2x to the power 5 so this comes out to be equal to 1 minus n x plus 40 x square minus 80 x cube plus 80 into x to the power 4 minus 32 into x to the power 5 this is the expansion of the expression 1 minus 2x whole to the power 5 now next we shall discuss general term in the expansion of a plus b whole to the power n the general term of the expansion is denoted by tr plus 1 and it is the r plus 1th term of the expansion of a plus b whole to the power n and we have tr plus 1 is equal to ncr into a to the power n minus r into b to the power r next we discuss the middle term if we have n is even then middle term is given by n upon 2 plus one-eth term and if we have n is odd then there will be two middle terms given by n plus 1 upon two-eth term and n plus 1 upon 2 plus one-eth term these are the two middle terms when n is odd let's consider the same expression 1 minus 2x whole to the power 5 let's try and find out the second term of this binomial expansion for this we will take r equal to 1 so we get t2 that is the second term is equal to ncr now here we have n is 5 5 c1 since r is equal to 1 into a to the power n minus r a is 1 so 1 to the power 5 minus 1 into b which is minus 2x to the power r that is 1 so this is equal to 5 into minus 2x which is equal to minus 10x that is the second term of the expansion of this expression is minus 10x now here since we have n is equal to 5 which is odd so the middle terms of the expansion are given by n plus 1 that is 5 plus 1 upon two-eth term and n plus 1 upon 2 plus one-eth term that is the third term and the fourth term of the expansion are the middle terms we have already expanded the expression 1 minus 2x whole to the power 5 which is equal to 1 minus 10x plus 40x square minus 80xq plus 80 into x to the power 4 minus 32 into x to the power 5 now the third term in this expansion is this and the fourth term is this so these two are the middle terms of the expansion so this is how we find the middle terms of the expansion of a given expression this completes the session hope you have understood the concept of binomial theorem