====== Question 3 & 4 Exercise 6.1 ======
Solutions of Question 3 & 4 of Exercise 6.1 of Unit 06: 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.

=====Question 3(i)=====
Prove that $\dfrac{1}{6 !}+\dfrac{2}{7 !}+\dfrac{3}{8 !}=\dfrac{75}{8 !}$
====Solution====
We are taking the L.H.S of the above given equation.
\begin{align}\dfrac{1}{6 !}+\dfrac{2}{7 !}+\dfrac{3}{8 !}&=\dfrac{1}{6 !}+\dfrac{2}{7.6 !}+\dfrac{3}{8.7 .6 !} \\
& =\dfrac{56+16+3}{8 !}\\
&=\dfrac{75}{8 !}\end{align}

=====Question 3(ii)=====
Prove that $\dfrac{(n+5) !}{(n+3) !}=n^2+9 n+20$
====Solution====
We are taking the L.H.S of the above given equation.
\begin{align}\dfrac{(n+5) !}{(n+3) !}&=\dfrac{(n+5)(n+4)(n+3) !}{(n+3) !} \\
& =(n+5)(n+4)\\
&=n^2+9 n+20\end{align}

=====Question 4(i)=====
Find the value of $n$, when $\dfrac{n(n !)}{(n-5) !}=\dfrac{12(n !)}{(n-4) !}$
====Solution====
We are given:
\begin{align}\dfrac{n(n !)}{(n-5) !}&=\dfrac{12(n !)}{(n-4) !} \\
\Rightarrow \dfrac{n}{(n-5) !}&=\dfrac{12}{(n-4)(n-5) !}  \\
\Rightarrow n&=\dfrac{12}{(n-4)} \\
\Rightarrow n(n-4)&=12 \\
\rightarrow n^2-4 n-12&=0 \\
\rightarrow n^2+2 n-6 n \quad 12&=0 \\
\Rightarrow n(n+2)-6(n+2)&=0 \\
\Rightarrow(n-6)(n+2)&=0 \\
\Rightarrow \text { Either. } n&=-2 \text { or } n=6\end{align}
$n$ can not b negative, therefore $n=6$.

=====Question 4(ii)=====
Find the value of $n$, when $\dfrac{n !}{(n-4) !}: \dfrac{(n-1) !}{(n-4) !}=9: 1$
====Solution====
We are given:
\begin{align} \dfrac{n !}{(n-4) !}: \dfrac{(n-1) !}{(n-4) !}&=9: 1 \\
\Rightarrow \dfrac{n !}{(n-4) !} \times \dfrac{(n-4) !}{(n-1) !}&=9 \\
\Rightarrow \dfrac{n !}{(n-1) !}&=9 \\
\Rightarrow \dfrac{n(n-1) !}{(n-1) !}&=9 \\
\Rightarrow n&=9 \end{align}







====Go To====
<text align="left"><btn type="primary">[[math-11-kpk:sol:unit06:ex6-1-p1 |< Question 1 & 2 ]]</btn></text>
<text align="right"><btn type="success">[[math-11-kpk:sol:unit06:ex6-1-p3|Question 5 >]]</btn></text>