Question

Binomial Theorem Ques!

Answer 1

find coefficient of \[x^n\] in \[(1+x+x^2)^n\]

Answer 2

\(((1+x)+x^2)^n=\sum _{k=0}^n\binom{n}{k} (1+x)^{(n-k)}x^{2k}\)

Answer 3

\(\large ((1+x)+x^2)^n=\\\large \sum _{k=0}^n\binom{n}{k} \left ({\sum _{m=0}^{n-k}\binom{n-k}{m}x^{(m)}} \right )x^{2k}\)

Answer 4

first term when k=0 and m=n
so
\( \binom{n}{0} \binom{n}{0}x^{(n)}=x^n \)

Answer 5

second term when
k=n/4
m=n/2
\(\binom{n}{\frac{n}{4}} \binom{\frac{4n-n}{4}}{\frac{n}{2}}x^{(n)}\)

Answer 6

so we would have
\(\Huge (\binom{n}{\frac{n}{4}} \binom{\frac{4n-n}{4}}{\frac{n}{2}}+1) x^n\)

Answer 7

i dont uderstand the notation you wrote under brackets if that means thsi then i gues it it correct
\[1+ \frac{ n(n-1) }{ 1!}+\frac{ n(n-1)(n-2)(n-3) }{ (2!)^2 }...............\]

Answer 8

yes , it mean binomial coefficient

Answer 9

can you explain this notation?i Dont know this

Answer 10

ok , could u please write what is binomial theorem ?

Answer 11

\[(a+b)^n=\sum_{r=0}^{n} c_{r}^{n} a ^{n-r} \times b^r\]

Answer 12

oh ok :P
so \( \binom{n}{r}=C_r^n\)

Answer 13

i used different notations only

Answer 14

ohhk ty man life saver.

Answer 15

girl :P

Answer 16

lol

Answer 17

so it would be
\(1+C_{\frac{n}{4}}^n +C_{\frac{n}{2}}^\frac{4n-n}{4} \)

Answer 18

this girl is not a ordinary girl :P

Answer 19

oh sry i made a typo
\(1+C_{\frac{n}{4}}^n C_{\frac{n}{2}}^\frac{4n-n}{4}\)

Answer 20

seriously.
too Over powerful.

Answer 21

:P

Answer 22

:3