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Mathematics
Binomial coefficient series
The value of the expression ${}^{n+1}{{C}_{2}}+2\left[ {}^{2}{{C}_{2}}+{}^{3}{{C}_{2}}+{}^{4}{{C}_{2}}+...+{}^{n}{{C}_{2}} \right]$ is $\dfrac{n\left( n+k \right)\left( pn+m \right)}{h}$. Find $k+m+p+h$.

Mathematics
Binomial coefficient series
A coin is tossed $n$ times. The chance that the head will present itself an odd number of times is $\dfrac{1}{k}$ . find the value of $k$

Mathematics
Binomial coefficient series
The value of ${}^{n}{{C}_{0}}{}^{n}{{C}_{r}}+{}^{n}{{C}_{1}}{}^{n}{{C}_{r+1}}+{}^{n}{{C}_{2}}{}^{n}{{C}_{r+2}}+....+{}^{n}{{C}_{n-r}}{}^{n}{{C}_{n}}$ is equal to?
(a) ${}^{2n}{{C}_{r}}$
(b) ${}^{2n}{{C}_{n+r}}$
(c) ${}^{2n}{{C}_{r-1}}$
(d) ${}^{2n}{{C}_{r+1}}$

Mathematics
Binomial coefficient series
If we have an expression as ${{\left( 1+x \right)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+...+{{C}_{n}}{{x}^{n}}$, then prove the following: ${{C}_{0}}{{C}_{r}}+{{C}_{1}}{{C}_{r+1}}+...+{{C}_{n-r}}{{C}_{n}}=\dfrac{\left( 2n \right)!}{\left( n+r \right)!\left( n-r \right)!}$

Mathematics
Binomial coefficient series
How do you find the binomial coefficient of $^{12}{{C}_{5}}$ ?
Mathematics
Binomial coefficient series
How do you use the binomial series to expand ${{\left( 1+2x \right)}^{5}}$?

Mathematics
Binomial coefficient series
How would you use the Maclaurin series for ${{e}^{-x}}$ to calculate ${{e}^{0.1}}$?
Mathematics
Binomial coefficient series
The value of $^{n}{{C}_{0}}^{n}{{C}_{2}}{{+}^{n}}{{C}_{1}}^{n}{{C}_{3}}{{+}^{n}}{{C}_{2}}^{n}{{C}_{4}}+.........{{+}^{n}}{{C}_{n-2}}^{n}{{C}_{n}}$ is equal to:
(a) $^{2n}{{C}_{n-2}}$
(b) $^{2n}{{C}_{n+1}}$
(c) $^{2n}{{C}_{n-1}}$
(d) None of these.

Mathematics
Binomial coefficient series
If we have an expression as $\alpha ={}^{m}{{C}_{2}}$, then ${}^{\alpha }{{C}_{2}}$ is equal to: -
(a) ${}^{m+1}{{C}_{4}}$
(b) ${}^{m-1}{{C}_{4}}$
(c) $3.{}^{m+2}{{C}_{4}}$
(d) $3.{}^{m+1}{{C}_{4}}$

Mathematics
Binomial coefficient series
The number of integral terms in the expansion of ${{\left( 2\sqrt{5}+\sqrt[6]{7} \right)}^{642}}$ are:
A. 105
B. 107
C. 321
D. 108

Mathematics
Binomial coefficient series
For any positive integers m, n (with $n\ge m$), let $\left( \begin{matrix} n \\ m \\ \end{matrix} \right)={}^{n}{{C}_{m}}$. Prove that $\left( \begin{matrix} n \\ m \\ \end{matrix} \right)+\left( \begin{matrix} n-1 \\ m \\ \end{matrix} \right)+\left( \begin{matrix} n-2 \\ m \\ \end{matrix} \right)+......+\left( \begin{matrix} m \\ m \\ \end{matrix} \right)=\left( \begin{matrix} n+1 \\ m+1 \\ \end{matrix} \right)$?

Mathematics
Binomial coefficient series
Find the value of ${(1.01)^5}$ correct to $5$ decimal places.

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