Question

Using Binomial theorem, evaluate $(101{)}^4$.

Answer 1

Given: $(101{)}^4=(100+1{)}^4$
We know that $(a+b{)}^n{=}^n{C}_0{a}^n{+}^n{C}_1{a}^{n-1}b+...{+}^n{C}_{n-1}a{b}^{n-1}{+}^n{C}_n{b}^n$
Putting $a=100,b=1,n=4$
$(100+1{)}^4{=}^4{C}_0(100{)}^4{+}^4{C}_1\left(100{)}^3\right(1\left){+}^4{C}_2\right(100{)}^2\left(1{)}^2{+}^4{C}_3\right(100\left)\right(1{)}^3{+}^4{C}_4(1{)}^4$

$(100+1{)}^4=(1\left)\right(100000000)+4(1000000\left)\right(1)+\frac{4\times 3}{2}(10000)+4(100)+1$

$(100+1{)}^4=100000000+4000000+60000+400+1$

$(100+1{)}^4=104060401$
Hence, $(101{)}^4=104060401$