====== Question 10 Exercise 7.2 ======
Solutions of Question 10 of Exercise 7.2 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Q10 Show that the sum of binomial coefticients of order $n=2 ;$. Also prove the sun of the odd hinomial coneficients=suin of even binomial cosficient $s=2^{n-1}$.
Solution: We know that
$$
\left.(1+x)^n=\left(\begin{array}{l}
n \\
\vdots
\end{array}\right)+\left(\begin{array}{l}
m \\
1
\end{array}\right) x+\left(\begin{array}{l}
n \\
2
\end{array}\right) x^2-\ldots+i_n^*\right) x^n \cdot
$$

Putting $x=1$ in the above equation, we have $(1 \div 1)^n=\left(\begin{array}{l}n \\ 0\end{array}\right)+\left(\begin{array}{l}n \\ 1\end{array}\right)+\left(\begin{array}{l}n \\ 2\end{array}\right)$.
$$
\begin{aligned}
& \left(\begin{array}{l}
n \\
3
\end{array}\right)+\ldots+\left(\begin{array}{l}
n \\
n
\end{array}\right) \\
& 2^n=\left(\begin{array}{l}
n \\
0
\end{array}\right)+\left(\begin{array}{l}
n \\
1
\end{array}\right) \div\left(\begin{array}{l}
n \\
2
\end{array}\right)+\left(\begin{array}{l}
n \\
i
\end{array}\right)+\ldots+\left(\begin{array}{l}
n \\
n
\end{array}\right) .
\end{aligned}
$$
which shows that the sum of the :nefficiens is $?^n$.
Now we know that
$$
\begin{aligned}
& (1+x)^n=\left(\begin{array}{l}
n \\
0
\end{array}\right)+\left(\begin{array}{c}
n \\
1
\end{array}\right) x \cdot\left(\begin{array}{l}
n \\
2
\end{array}\right) x^2+1_3^1 x^2- \\
& \left(\begin{array}{l}
n \\
4
\end{array}\right) x^4+\ldots+\left(\begin{array}{l}
n \\
n
\end{array}{ }_1\right) x^n{ }^1+\left(\begin{array}{c}
n \\
n
\end{array}\right) 1^n \\
&
\end{aligned}
$$

If we put $x=-1$ in the above eyuation, we get
$$
\begin{aligned}
& 0=\left(\begin{array}{l}
n \\
1
\end{array}\right)-\left(\begin{array}{l}
n \\
1
\end{array}\right)+\left(\begin{array}{l}
n \\
2
\end{array}\right) x^2-\left(\begin{array}{l}
n \\
3
\end{array}\right)+\left(\begin{array}{l}
4 \\
4 \\
4
\end{array}\right) \\
& +\ddots^n,(-1)^{n-1} \ldots\left(n_n^{\prime \prime}(\cdots 1)^n\right. \\
&
\end{aligned}
$$

Vow we have two cases
Case- 1 If $n$ is caen then
$$
\begin{aligned}
& \left(\begin{array}{c}
5 \\
y
\end{array}\right) \cdot\left(\begin{array}{l}
n \\
\vdots
\end{array}\right)+\left(\begin{array}{l}
n \\
4
\end{array}\right)+\ldots \quad\left(\begin{array}{l}
n \\
\vdots
\end{array}\right)=\left(\begin{array}{l}
n \\
i
\end{array}\right)+\left(\begin{array}{l}
n \\
7
\end{array}\right) \\
& \left.\left(\frac{n}{5}\right)-\ldots . .+n_n^n 1\right) \\
&
\end{aligned}
$$
and hence the sum of even and add coeflicienis are equat.
Case-2 If $n$ is odd then
$$
\begin{aligned}
& \left(\begin{array}{l}
e^2 \\
3
\end{array}\right)+\left(\begin{array}{l}
4 \\
5
\end{array}\right)+\ldots+\left(\begin{array}{l}
a \\
a
\end{array}\right) \\
&
\end{aligned}
$$
and hence the sum of even and odd chefficients are cyual.

Nins we have shown that
uncomplete question ./.;;./;;....





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