Permutations And Combinations

We want at least 2 letters in wrong envelopes. This is the inclusion of all arrangements where 2 or more letters are deranged.
The number of such arrangements is: Where is the number of derangements of items.

We are given the equation: Factorize . We must distribute 3 powers of 2 and 1 power of 3 among the three variables. Each distribution of powers among corresponds to a combination of: Number of positive integer solutions:

With repetition allowed, number of -length permutations:
So total permutations for to is: But options suggest indexing from to : i.e., However, question says “not more than ” and starts from 1 ⇒ correct is:

Given series: But So,

To count trailing zeros in , use:

We want the smallest for which sum equals 1000. Try :




Sum =
So,

Given: Use the property: Also: Solving equations (1) and (2) simultaneously gives: Hence:

“MULTIPLE” has 8 letters. Vowels = U, I, E.
If positions of vowels are fixed, arrange only the 5 consonants: M, L, T, P, L
L is repeated twice. So number of arrangements =
But there’s a catch: positions of vowels are fixed among the original arrangement.
Assuming vowel positions are fixed, the 5 consonants are being permuted in their slots.
Correct interpretation: total = (as MULTIPLE has repeated L and total 8 letters, 3 vowel slots fixed)

An injection means one-to-one mapping. Each element in must go to a distinct element in .
So number of injections from to is the number of ways to assign distinct elements to available spots (without repetition):
This is permutation:

We have 4 distinct digits: 2, 3, 5, 7. Numbers can be 1-digit, 2-digit, 3-digit, or 4-digit. The question implies "sum of all possible numbers" meaning numbers using all 4 digits without repetition (4-digit numbers). If it means sum of all numbers of any length, the problem is different. Let's assume it means sum of all 4-digit numbers formed by these digits. Sum of digits = . Number of digits . The number of permutations (4-digit numbers) is . In these 24 numbers, each digit (2, 3, 5, 7) will appear in each position (units, tens, hundreds, thousands) an equal number of times. Number of times each digit appears in a specific position = times. Sum of digits in the units place = . Sum of digits in the tens place = . Contribution = . Sum of digits in the hundreds place = . Contribution = . Sum of digits in the thousands place = . Contribution = . Total sum = . . Let's check the options. % Option (a) . (Incorrect for 4-digit sum) If n digits, sum is (sum of digits) . Here, sum of digits = 17. . String of ones = 1111. Sum = . Now evaluate option (b): . . . This matches my calculated sum. So option (b) is correct. The formula in options (a) and (c) gives a string of k ones (e.g., ). Option (a) is not right. The general formula for sum of all n-digit numbers formed by n distinct non-zero digits is: Sum = (Sum of the n digits) ((n-1)!) (111...1, n times). Here, Sum = . Option (b) evaluates to 113322.

We know the formula for permutations: . The given recurrence relation is . Let's expand the terms: . . Substitute these into the equation: We can write and . Assuming and , divide by : Multiply by : . This matches option (d). This recurrence relation is a known identity for permutations.

The task is to form words with 2 different vowels and 2 different consonants. Number of vowels = 5. Number of consonants = 21. Step 1: Select 2 different vowels from 5 vowels. Number of ways = . Step 2: Select 2 different consonants from 21 consonants. Number of ways = . Step 3: Combine these selections. Number of ways to select the group of 4 letters (2 different vowels, 2 different consonants) = (Ways to select vowels) (Ways to select consonants) = . Step 4: Arrange these 4 selected distinct letters to form a word. The 4 selected letters are all different (2 different vowels, 2 different consonants, and vowels are different from consonants). Number of ways to arrange 4 distinct letters = . . Total number of words that can be formed = (Number of ways to select the group of 4 letters) (Number of ways to arrange these 4 letters) Total words = . This matches option (d).

A chord is determined by 2 distinct points on the circle. So total number of chords is:

The word "KANGAROO" has 8 letters: K, A, N, G, A, R, O, O. The total number of arrangements without any restrictions is given by: Now, we calculate the number of arrangements where the A's appear together. We treat the A's as a single unit, so the arrangement becomes K, (AA), N, G, R, O, O. The total number of such arrangements is: Thus, the number of arrangements where the A's do not appear together is: Therefore, the correct answer is 7560.

VOWEL has 5 letters. Vowels = O, E ⇒ keep them together as a block. Treat OE as one block → blocks = [OE, V, W, L] = 4 items ⇒ ways. Within the block OE: 2! ways to arrange.

To find the number of 8-digit odd numbers, consider the following: - The first digit must be any digit from 1 to 9 (9 options). - The last digit must be an odd number, i.e., 1, 3, 5, 7, or 9 (5 options). - The remaining 6 digits can be any digit from 0 to 9 (10 options each). Thus, the total number of 8-digit odd numbers is:

We are given that the number is a 4-digit number starting with 4 and ending with either 0 or 5. The choices for the second and third digits are any digits from 0 to 9 (10 options for each). Thus, the total number of such 4-digit numbers is:

We know that stands for the binomial coefficients: The sum of all binomial coefficients for a fixed is known to be: Thus, the sum of the given series is: After applying the given transformation, we simplify the expression, leading to the final result: Thus, the correct answer is .

"ARRANGEMENT" has 11 letters. Frequency: A-1, R-2, N-2, G-1, E-2, M-1, T-1. Total arrangements with repeated letters: Now count arrangements where the two Es are adjacent: treat "EE" as one unit. We now arrange 10 units: Subtract: Answer is simplified as

Step 1: Identify valid combinations of questions.
Let be the number of questions from Section A, and from Section B. Then: Valid combinations:
Step 2: Calculate combinations for each case.
Step 3: Add all possible ways.

Step 1: Use integer partition method for dice We are looking for the number of non-negative integer solutions to: where each and . Convert to: Now number of integer solutions of is:

We are asked to form four-digit numbers using the digits with all digits distinct. Step 1: Calculating , the total number of four-digit numbers The number of ways to choose the first digit is 7 (since it can be any digit from 1 to 7). The number of ways to choose the second digit is 6 (since one digit is already used). The number of ways to choose the third digit is 5, and the number of ways to choose the fourth digit is 4. Thus, the total number of numbers is: Step 2: Calculating , the number of numbers greater than 3400 For numbers greater than 3400, the first digit must be 3, 4, 5, 6, or 7. - If the first digit is 3, the second digit must be 4 or greater, which leaves us with 4 choices for the second digit. The remaining 2 digits can be chosen in ways. Thus, the number of such numbers is . - If the first digit is 4, the second digit can be any of the remaining digits (6 choices), and the third and fourth digits can be chosen in ways. Thus, the number of such numbers is . - For first digits 5, 6, and 7, the number of possible numbers is similar to the case when the first digit is 4. Thus, the total number of numbers greater than 3400 is: Step 3: Finding the ratio The ratio is: Thus, the ratio is .

We are given that there are 6 different novels and 3 different poetry books. We need to select 4 novels and 1 poetry book, and arrange them in a row on a shelf with the condition that the poetry book is always in the middle. Step 1: Arranging the poetry book in the middle Since the poetry book must always be in the middle, we have only 1 choice for the position of the poetry book. There is only 1 position for the poetry book in the middle of the 5 positions on the shelf. Step 2: Selecting and arranging the novels We need to select 4 novels from the 6 available novels. The number of ways to choose 4 novels from 6 is given by the combination formula: After selecting the 4 novels, we can arrange them in the 4 remaining positions. The number of ways to arrange 4 novels is , which is: Step 3: Selecting the poetry book Since there are 3 different poetry books, we can choose any 1 of them in 3 ways. Step 4: Calculating the total number of arrangements The total number of arrangements is given by: Thus, the total number of possible arrangements is .

We are tasked with forming a five-digit number divisible by 3 using the digits 0, 1, 2, 3, 4, and 5 without repetition. A number is divisible by 3 if the sum of its digits is divisible by 3. Step 1: Total sum of the digits The sum of the digits 0, 1, 2, 3, 4, and 5 is: Since the total sum of the digits is 15, which is divisible by 3, the sum of the digits of any five-digit number formed from these digits will also be divisible by 3, provided that we leave out one of the digits. Step 2: Choosing one digit to leave out We can leave out any of the six digits (0, 1, 2, 3, 4, 5). If we leave out a digit, the sum of the remaining digits will still be divisible by 3. We need to calculate the number of five-digit numbers that can be formed with the remaining digits. Step 3: Counting the number of five-digit numbers We must choose 5 digits from the 6 available digits. We have 6 choices for the digit to leave out. For each selection of 5 digits, the number of ways to arrange them is given by the number of permutations of 5 digits. However, the first digit cannot be 0, so we need to adjust the counting. - If 0 is not selected, all 5 digits can be arranged in ways. - If 0 is selected, the first digit cannot be 0, so we must select the first digit from the remaining 4 digits (1, 2, 3, 4, or 5) and arrange the other 4 digits. Step 4: Calculating the number of arrangements - If 0 is not selected, we have ways. - If 0 is selected, the number of valid arrangements is . Thus, the total number of arrangements is: Thus, the total number of ways to form a five-digit number divisible by 3 is .

Step 1: Understand the problem statement.
We need to form ordered pairs from the first 50 natural numbers (i.e., from the set ) such that .
Step 2: Relate the problem to combinations.
If we select any two distinct numbers from the set of 50 natural numbers, there is only one way to assign them to and such that . Specifically, the larger of the two chosen numbers will be , and the smaller will be .
For example, if we choose the numbers 10 and 25, then must be 25 and must be 10 to satisfy .
Therefore, the problem reduces to finding the number of ways to choose 2 distinct numbers from a set of 50 numbers. This is a combination problem.
Step 3: Calculate the number of combinations.
The number of ways to choose items from a set of distinct items is given by the combination formula:
In this case, (total natural numbers) and (numbers to choose for the pair).
So, there are 1225 ways to form such ordered pairs.
The final answer is .

The word `AGAIN' has letters A, A, G, I, N (5 letters, two A’s identical). Total permutations: Arrange in dictionary (alphabetical) order: A, G, I, N. Count permutations by first letter:
- A: Remaining letters A, G, I, N (4 letters, one A repeated):
- G: Remaining letters A, A, I, N:
- I: Remaining letters A, A, G, N:
- N: Remaining letters A, A, G, I:
Cumulative count: A (1–12), G (13–24), I (25–36), N (37–48). The word starts with N (since ). List words starting with N:
- NA: Remaining A, G, I: (NAAGI, NAAIG, NAGAI, NAGIA, NAIGA, NAIAG)
- NAA: Words are NAAIG, NAAGI (2 permutations)
- NAAG: Words are NAAGI, NAAGG (but NAAGG is invalid)
- NAAI: Words are NAAIG, NAAIA (but NAAIA is invalid)
Order: NAAIG, NAAGI, NAGAI, NAGIA, NAIGA, NAIAG. The is NAAGI, and the is NAAIG. Option (3) is correct. Options (1), (2), and (4) are not the word.

Step 1: Choose the other two digits
Since and must both appear exactly once, we select the other two digits from . Number of ways to choose these two digits: Step 2: Arrange the four chosen digits
Each chosen set of 4 digits (which always includes 3 and 7 plus the chosen two) can be permuted in: ways.
Step 3: Check leading digit restrictions
Since zero is not in the digit set, there is no restriction on the leading digit.
Step 4: Calculate total number of 4-digit numbers
Final answer: