====== Question 12 Exercise 6.2 ======
Solutions of Question 12 of Exercise 6.2 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 12(i)=====
How many different word can be formed from the letters "BOOKWORM" if the letters are taken all at a time?
====Solution====
BOOKWORM\\
The total number of letters in word BOOKWORM are $8.$

$n=8$ out of which three are $\mathrm{O}$, 

so $m_1=3$..

Thus total number of different words using all at a time are:
\begin{align}
 \left(\begin{array}{c}
n \\
m 1
\end{array}\right)&=\left(\begin{array}{l}
8 \\
3
\end{array}\right) \\
& =\dfrac{8 !}{3 !}\\
&=\dfrac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 !}{3 !}\\
&=6,720 \end{align}

=====Question 12(ii)=====
How many different word can be formed from the letters "BOOKKEEPER" if the letters are taken all at a time?
====Solution====
BOOKKEEPER\\
The total number of letters in $\mathrm{BOOK}$ KEEPER are ten.

$n=10$, out of which two are $\mathrm{O}$,

so $m_1=2$, three are $\mathrm{E}$,

so $m_2=3$, two are $\mathrm{K}$, 

so $m_3=2$. Thus the total number of different words are:
\begin{align}
\left(\begin{array}{c}
n \\
m_1, m_2, m_3
\end{array}\right)&=\left(\begin{array}{c}
10 \\
2,3.2
\end{array}\right) \\
& =\dfrac{10 !}{2 ! \cdot 3 ! \cdot 2 !}\\
&=151,200 \end{align}

=====Question 12(iii)=====
How many different word can be formed from the letters "ABBOTTABAD" if the letters are taken all at a time?
====Solution====
ABBOTABAD\\
Total number of letters are ten, so $n=10$ out of which three are $\mathrm{A}$,

so $m_1=3$, three are $B$, so $m_2=3$, and two are $T$,

so $m_3=2$.

Thus the total number of different words formed are:
\begin{align}
\left(\begin{array}{c}
n \\
m_1, m_2, m_3
\end{array}\right)&=\left(\begin{array}{c}
10 \\
3,3,2
\end{array}\right) \\
& =\dfrac{10 !}{3 ! \cdot 3 ! \cdot 2 !}\\
&=50,400 \end{align}

=====Question 12(iv)=====
How many different word can be formed from the letters "LETTER" if the letters are taken all at a time?
====Solution====
LETTER\\
The total number of letters in letter are six. 

so, $n=6$ out of which two are t,

so $m_1=2$ and two are e, so $m_2=2$.

Thus the total number of different words formed are:
\begin{align}\left(\begin{array}{c}
n \\
m_1, m_2
\end{array}\right)&=\left(\begin{array}{c}
6 \\
2,2
\end{array}\right)\\&=\dfrac{6 !}{2 ! \cdot 2 !}\\
&=180 \end{align}




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