Question

Determine the most efficient way to use the Binomial Theorem to show the following. (11)^4= 14641

A)Write 11=5+6 and expand.
B) Write 11= 10+1 and expand.
C) Write 11= 3+3+2 and expand.
D)Write 11= 4+4+3 and expand.

Answer 1

Write (1+10) and expand. As you will have nCr *(10)^(n-r)
Where r index n count ( less terms)

Answer 2

The formula for binomial theorem is

`(a+b)^n=( \left \ {{n} \atop {0}} \right.)a^(n)+( \left \ {{n} \atop {1}} \right.)a^(n-1)b+( \left \ {{n} \atop {0}} \right.)a^(n-2)b^2+...+( \left \ {{n} \atop {n}} \right.)b^n`

Now this shall be very easy if the value of a = 1

The formula shall become

`(1+b)^n=( \left \ {{n} \atop {0}} \right.)1^(n)+( \left \ {{n} \atop {1}} \right.)1^(n-1)b+( \left \ {{n} \atop {0}} \right.)1^(n-2)b^2+...+( \left \ {{n} \atop {n}} \right.)b^n`

Which shall be

`(1+b)^n=( \left \ {{n} \atop {0}} \right.)+( \left \ {{n} \atop {1}} \right.)b+( \left \ {{n} \atop {0}} \right.)b^2+...+( \left \ {{n} \atop {n}} \right.)b^n`

So to find 11^4

We must break it as 1 + 10.

Option B) is the right answer.