Permutation and combination practice · Mixed difficulty

Option D is correct because completing the square leads to an equation of a circle, and there are 8 integer pairs that satisfy it.

Option A (360) is correct because it accounts for the numbers starting with zero and subtracts them.

Option A (35) is correct because the '+' signs create 7 spaces where the '-' signs can be placed, and we need to choose 4 of those spaces.

Option C (96) is correct because it counts all the words that would appear before "COCHIN" in an alphabetical listing.

Option D (3999960) is correct because it applies the formula for the sum of all numbers formed by given digits without repetition.

Option B ($\frac{{9!}}{{2!2!}}$) is correct because it treats the three E's as one unit and accounts for the repetitions of C and N.

Option A (666) is correct because it represents the number of integers between 2 and 2000 that are not divisible by 2 or 3, calculated using the principle of inclusion-exclusion.

Option D (None of these) is correct because the calculated number of positive integral solutions is 1875, which is not among the given options.

Option D (15) is correct because there are 5 numbers with digits 9,9,9,9,7 and 10 numbers with digits 9,9,9,8,8, summing to 15.

Option B (7) is correct because it satisfies the condition that ${ }^n C_4,{ }^n C_5$ and ${ }^n C_6$ are in A.P., leading to the solution n=7 or n=14.

Option A (FATHER) is correct because when the words are arranged alphabetically, FATHER is the 261st word.

Option D (5) is correct because the number of different garlands is calculated using the formula for circular permutations with identical objects, accounting for reflections: (6!/(3!3!))/2 + cases where arrangement is symmetrical.

Option B (4960) is correct because it subtracts the number of four-digit numbers without repetition from the total number of four-digit numbers.

Option C (10) is correct because it represents the number of teams that satisfy the given condition of 45 matches played.

The correct answer is A (24) because it considers arranging U, CC, and E, and then placing the three S's in the available spaces such that no two S's are together.

Option A (5 !) is correct because the number of ways to arrange n persons around a circle is (n-1)!, so for 6 persons, it is (6-1)! = 5!.

Option A (14) is correct because the sum of the digits of N, which satisfies the given condition, is 14.

Option C (${ }^{17} {C}_7$) is correct because it represents the number of ways to distribute identical objects among persons without any restrictions using stars and bars method.

Option D (91) is correct because it is the total number of possible outcomes minus the number of outcomes where no dice shows 5.

Option D (626) is correct because it accounts for all possible combinations of 4 letters from the word 'MORADABAD', considering repetitions.

Option C (72) is correct because arranging the consonants and then placing the vowels in the gaps ensures that no two vowels are adjacent.

Option B (35) is correct because it represents the number of ways to distribute 16 identical items among 4 people, with each person receiving at least 3 items.

Option B (60) is correct because each digit appears in the unit's place $3!$ times, and the sum of the digits is $1+2+3+4=10$, so the sum of digits in the unit's place is $3! \times 10 = 60$.

Option C ($2^2 \cdot 3^3 \cdot 7^2$) is correct because it represents the total count of five-digit numbers with distinct digits where the middle digit is the largest, calculated using permutations and summation.

Option A (210) is correct because it calculates the total number of different tickets required for travel between any two stations.

Option A (1279) is correct because it represents the total number of factors excluding 1 and the expression itself, calculated using the exponents of the prime factors.

The correct answer is D (74) because it represents the rank of the word RAHUL in a dictionary formed by the letters A, H, L, R, U.

Option A (264) is correct because it represents the number of ways to place 6 letters into wrong envelopes, considering 5 letters are misplaced.

Option C (264) is correct because it represents the total number of ways to form a team with at least 4 bowlers, calculated using combinations.

Option B (60) is correct because we subtract the cases where all letters are posted in the same box from the total possible ways of posting the letters.