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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