Find the number of words with or without meaning which can be made using all the letters of the word $AGAIN$. If these words are written as in a dictionary, what will be the $50^{\text {th }}$ word?
Find the number of words with or without meaning which can be made using all the letters of the word $AGAIN$. If these words are written as in a dictionary, what will be the $50^{\text {th }}$ word?
There are $5$ letters in the word $AGAIN$, in which $A$ appears $2$ times. Therefore, the required number of words $=\frac{5 !}{2 !}=60$
To get the number of words starting with $A$, we fix the letter $A$ at the extreme left position, we then rearrange the remaining $4$ letters taken all at a time. There will be as many arrangements of these $4$ letters taken $4$ at a time as there are permutations of $4$ different things taken $4$ at a time. Hence, the number of words starting with $A=4 !=24 .$ Then, starting with $G$, the number of words $=\frac{4 !}{2 !}=12$ as after placing $G$ at the extreme left position, we are left with the letters $A , A , I$ and $N$. Similarly, there are $12$ words starting with the next letter $I$. Total number of words so far obtained $=24+12+12=48$
The $49^{\text {th }}$ word is $NAAGI$. The $50^{\text {th }}$ word is $NAAIG$.
Similar Questions
Determine $n$ if
$^{2 n} C_{3}:\,^{n} C_{3}=12: 1$
Determine $n$ if
$^{2 n} C_{3}:\,^{n} C_{3}=12: 1$
In a shop there are five types of ice-creams available. A child buys six ice-creams.
Statement $-1 :$ The number of different ways the child can buy the six ice-creams is $^{10}C_5.$
Statement $-2 :$ The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging $6 \,A's$ and $4 \,B's$ in a row.
In a shop there are five types of ice-creams available. A child buys six ice-creams.
Statement $-1 :$ The number of different ways the child can buy the six ice-creams is $^{10}C_5.$
Statement $-2 :$ The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging $6 \,A's$ and $4 \,B's$ in a row.
- [AIEEE 2008]
Suppose Anil's mother wants to give $5$ whole fruits to Anil from a basket of $7$ red apples, $5$ white apples and $8$ oranges. If in the selected $5$ fruits, at least $2$ orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer $5$ fruits to Anil is $........$
Suppose Anil's mother wants to give $5$ whole fruits to Anil from a basket of $7$ red apples, $5$ white apples and $8$ oranges. If in the selected $5$ fruits, at least $2$ orange, at least one red apple and at least one white apple must be given, then the number of ways, Anil's mother can offer $5$ fruits to Anil is $........$
- [JEE MAIN 2023]
Determine the number of $5$ card combinations out of a deck of $52$ cards if there is exactly one ace in each combination.
Determine the number of $5$ card combinations out of a deck of $52$ cards if there is exactly one ace in each combination.
A father with $8$ children takes them $3$ at a time to the Zoological gardens, as often as he can without taking the same $3$ children together more than once. The number of times each child will go to the garden is
A father with $8$ children takes them $3$ at a time to the Zoological gardens, as often as he can without taking the same $3$ children together more than once. The number of times each child will go to the garden is