
Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements.
Do vowels never occur together?
A. $36000$
B. $4320$
C. $10080$
D. $40230$
Answer
511.8k+ views
Hint: The number of permutations of n objects, where p objects are of the same kind and rest are all different$ = \dfrac{{n!}}{{p!}}$.Number of permutation $ = $total number of permutation – number of permutation of all vowels occurring together
Complete step by step solution:
The given world is INDEPENDENCE.
Here, $E$ repeats$4$ times, $N$ repeats $3$ times, $D$ repeats $2$ times and $E\,\,and\,\,I$ are vowels,
As they occur together, we treat them as single object $EEEI$
Now, Arranging sets of $5$ vowels as $EEEEI$,$IEEEE,EEIEE$ and so on.
In $EEEEI;$ there are $4E$ where letter E is repeating,
Now, we use the formula:\[\dfrac{{n!}}{{{P_1}!{P_2}!{P_3}!}}\].
Total number of letters $(n) = 4 + 1 = 5$
Here, \[{P_1} = 4\]
Total arrangements $ = \dfrac{{5!}}{{4!}}$
\[ = \dfrac{{5 \times 4!}}{{4!}}\]
$ = 5$
Numbers that we need to arrange $ = 7 + 1 = 8$
Here, $N = 3\,\,and\,\,D = 2$
Since letters are repeating,
Now, we use the formula $\dfrac{{n!}}{{{P_1}{P_2}{P_3}}}$
Total letters $(n) = 8$
and ${P_1} = 3$, for letter N
${P_2} = 2$, for letter D
Total arrangement $ = \dfrac{{8!}}{{3!2!}}$
$ = 56 \times 6 \times 10$
$
= 336 \times 10 \\
= 3360 \\
$
Thus, number of arrangements $ = $total arrangements of vowels \[ \times \]total arrangements of letter
$
= 5 \times 3360 \\
= 16800 \\
$
Now, in words INDEPENDENCE
There are $3 - N({P_1})$
$4 - E({P_2})$
$2 - D({P_3})$
Total letters $ = 12$
So, $n = 12$
As letters are repeating, we use the formula as given below,
$\dfrac{{n!}}{{{P_1}!{P_2}!{P_3}!}}$
Total arrangements $ = \dfrac{{12!}}{{3!4!2!}}$
$ = \dfrac{{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4!}}{{4!\,3 \times 2 \times 1 \times 2}}$
$ = 1663200$
Number of arrangements where vowels never occur together $ = $total number of arrangement – Number of arrangements when all vowels occur together
$ = 1663200 - 16800$
$ = 1646400$
Note: In these types of questions make sure no combination of alphabets is missed and the cases of repetition too are to be solved only once.
Complete step by step solution:
The given world is INDEPENDENCE.
Here, $E$ repeats$4$ times, $N$ repeats $3$ times, $D$ repeats $2$ times and $E\,\,and\,\,I$ are vowels,
As they occur together, we treat them as single object $EEEI$
Now, Arranging sets of $5$ vowels as $EEEEI$,$IEEEE,EEIEE$ and so on.
In $EEEEI;$ there are $4E$ where letter E is repeating,
Now, we use the formula:\[\dfrac{{n!}}{{{P_1}!{P_2}!{P_3}!}}\].
Total number of letters $(n) = 4 + 1 = 5$
Here, \[{P_1} = 4\]
Total arrangements $ = \dfrac{{5!}}{{4!}}$
\[ = \dfrac{{5 \times 4!}}{{4!}}\]
$ = 5$
Numbers that we need to arrange $ = 7 + 1 = 8$
Here, $N = 3\,\,and\,\,D = 2$
Since letters are repeating,
Now, we use the formula $\dfrac{{n!}}{{{P_1}{P_2}{P_3}}}$
Total letters $(n) = 8$
and ${P_1} = 3$, for letter N
${P_2} = 2$, for letter D
Total arrangement $ = \dfrac{{8!}}{{3!2!}}$
$ = 56 \times 6 \times 10$
$
= 336 \times 10 \\
= 3360 \\
$
Thus, number of arrangements $ = $total arrangements of vowels \[ \times \]total arrangements of letter
$
= 5 \times 3360 \\
= 16800 \\
$
Now, in words INDEPENDENCE
There are $3 - N({P_1})$
$4 - E({P_2})$
$2 - D({P_3})$
Total letters $ = 12$
So, $n = 12$
As letters are repeating, we use the formula as given below,
$\dfrac{{n!}}{{{P_1}!{P_2}!{P_3}!}}$
Total arrangements $ = \dfrac{{12!}}{{3!4!2!}}$
$ = \dfrac{{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4!}}{{4!\,3 \times 2 \times 1 \times 2}}$
$ = 1663200$
Number of arrangements where vowels never occur together $ = $total number of arrangement – Number of arrangements when all vowels occur together
$ = 1663200 - 16800$
$ = 1646400$
Note: In these types of questions make sure no combination of alphabets is missed and the cases of repetition too are to be solved only once.
Recently Updated Pages
Master Class 12 Business Studies: Engaging Questions & Answers for Success

Master Class 12 English: Engaging Questions & Answers for Success

Master Class 12 Economics: Engaging Questions & Answers for Success

Master Class 12 Social Science: Engaging Questions & Answers for Success

Master Class 12 Maths: Engaging Questions & Answers for Success

Master Class 12 Chemistry: Engaging Questions & Answers for Success

Trending doubts
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Gautam Buddha was born in the year A581 BC B563 BC class 10 social science CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

What is the median of the first 10 natural numbers class 10 maths CBSE

Change the following sentences into negative and interrogative class 10 english CBSE
