Permutations

The number of three digit numbers in which 9 appears at hundred place only is Similarly, the number of three digit numbers in which 9 appears at ten or unit place only is

So, required number

If we arrange the letter of word ANIMAL in alphabetic order then, we get AAILMN, now to find the rank of word ANIMAL, (given)

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Now, similarly for word PERSON, if we arrange the letters in alphabetic order, we get ENOPRS, now to find the word having 107 ways,



So, ESORNP is required word.

To determine the number of 4
-digit numbers formed using the digits (with repetition allowed) that are divisible by 4, we proceed as follows: Step 1: Total number of 4
-digit numbers A 4
-digit number cannot start with . Therefore:
- The first digit (thousands place) can be or (4 choices).
- The second, third, and fourth digits (hundreds, tens, and units places) can be or (5 choices each). Thus, the total number of 4
-digit numbers is: Step 2: Divisibility by 4 A number is divisible by 4 if the last two digits form a number that is divisible by 4. Therefore, we need to count the number of valid 4
-digit numbers where the last two digits form a number divisible by 4. Step 3: Count valid last two digits The last two digits (tens and units places) must form a number divisible by 4. Using the digits , the possible valid pairs for the last two digits are: However, since the digits available are , the valid pairs are: There are 8 valid pairs for the last two digits. Step 4: Count valid 4
-digit numbers For each valid pair of last two digits:
- The first digit (thousands place) can be or (4 choices).
- The second digit (hundreds place) can be or (5 choices). Thus, for each valid pair of last two digits, there are: valid 4
-digit numbers. Since there are 8 valid pairs, the total number of 4
-digit numbers divisible by 4 is: Final Answer: