Hello folks , i wanted to discuss some really cool derivations and formulas related to binomial theorem . There is so much material on this topic that i will have to do a part 2 next week. For now , we will be focusing on proving one formula.
The Total number of terms in expansion of
is equal to or
where represents the power of the expression
and represents the numbers of terms .
Now sit back tight and get some popcorn . Let's begin with the proof !
proof: From Multinomial theorem the general term of the expansion can be written as
It is important to observe that number of terms in the expansion is equal to different sets of values of , which satisfies the above two conditions.
To calculate all the permutations . Let us denote all the possible values of the variables , and so on as the powers of .
We want the Sum of variables to be equal to . Which means we are trying to find coefficient of . Using formula of G.P
From the pattern
is nothing but . To find coeff. of put
Coefficient of is given by or
I hope you enjoyed this note . Thank you.