tell me all the properties of binomial theorem
Answers
The number of terms in the expansion is (n+1) which is one more than the index. The sum of the indices in x and y is n. The first and the last term being an and bn respectively. If nCx = nCy, then either x = y or x + y = n.
Answer:
I think this is the answer
Step-by-step explanation:
The number of terms in the expansion is (n+1) which is one more than the index.
The sum of the indices in x and y is n.
The first and the last term being an and bn respectively. If nCx = nCy, then either x = y or x + y = n
=> nCr = nCn-r = n!/r!(n-r)!
The general term in the expansion of (a + x)n is (r + 1)th term given as
Tr+1 = nCran-r + xr.
Some of the standard properties of binomial coefficients which should be remembered are:
C0 + C1 + C2 + ….. + Cn = 2n
C0 + C2 + C4 + ….. = C1 + C3 + C5 + ….. = = 2n-1
C02 + C12 + C22 + ….. + Cn2 = 2n Cn = (2n!)/ n!n!
Similarly the general term in the expansion of (x + a)n is given as
Tr+1 = nCr xn-r ar. The terms are considered from the beginning.
The binomial coefficient in the expansion of (a + x)n which are equidistant from the beginning and the end are equal i.e. nCr = nCn-r.
Another result that is applied in questions is nCr + nCr-1 = n+1Cr.
We can also replace mC0 by m+1C0 because numerical value of both is same i.e. 1. Similarly we can replace mCm by m+1Cm+1.