# 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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