# Properties of Newton's Binom Formula

## Related Calculator: Binomial Expansion Calculator

Properties for (a+b)^n are following:

1. Number of all summands in expansion equals n+1.
2. General member of expansion has form T_(k+1)=C_n^ka^(n-k)b^k, where k=0,1,2,...,n.
3. Coefficients that are equidistant from ends of expansion are equal.
4. Sum of all binomial coefficient is 2^n.
5. Sum of binomial coefficients on even positions equals sum of coefficients on odd positions.

Example 1. Find the greatest element in the expansion (a+b)^n, if sum of all binomial coefficients is 4096.

According to property 4, we have that 2^n=4096 or n=12.

Since 12 is even number than the greatest coefficient is near middle (i.e. 12/2=6-th) member. This coefficient equals C_(12)^6=(12!)/(6!(12-6)!)=(12!)/(6!6!)=(12*11*10*9*8*7*6!)/(6!6!)=924.

Example 2. In expansion of (sqrt(z)-2/(root(3)z))^15 find member that doens't contain z.

We can rewrite above expression as (z^(1/2)+(-2z^(-1/3)))^15. Here a=z^(1/2),b=-2z^(-1/3),n=15.

Now, using property 2, we have that T_(k+1)=C_(15)^k(z^(1/2))^(15-k)(-2z^(-1/3))^k=C_(15)^k(-2)^kz^((15-k)/2-k/3).

Member will not contain z, when power of z equals 0, i.e. (15-k)/2-k/3=0 or k=9.

Therefore, 10-th member doesn't contain z and it equals T_(9+1)=C_(15)^9(-2)^9=-2562560.

Example 3. Find the greatest binomial coefficient of expansion (n+1/n)^n, if product of forth member from the beginning and fourth member from the end equals 14400.

Fourth member from the beginning is T_4=T_(3+1)=C_n^3n^(n-3)*1/n^3, and fourth member from the end is T_((n+1-4)-1)=T_(n-2)=C_n^(n-3)n^3 1/(n^(n-3)).

Therefore, their product equals T_4T_(n-2)=C_n^3C_n^(n-3)=C_n^3C_n^3=(C_n^3)^2=14400. From this we have that C_n^3=120.

We can think a bit differently: from property 3 we have that coefficient are equal, coefficient near fourth member from beginning is C_n^3, therefore coefficient near fourth member from the end is also C_n^3; thus, their product is C_n^3*C_n^3=14400 or C_n^3=120.

Now, (n!)/((n-3)!3!)=120 or (n(n-1)(n-2)(n-3)!)/((n-3)!6)=120. This gives n(n-1)(n-2)=720=10*9*8, so n=10.

Therefore, the greatest coefficient is near 5-th member (10/2=5) and it equals C_(10)^5=(10!)/(5!5!)=252.