Permutation and combination practice · Easy

Option C (${9^n}$) is correct because each digit has 9 choices, and there are n digits.

Option B (3!) is correct because the ten's digit of the sum is 6, which is divisible by 3! (6).

Option D (36) is correct because it uses the principle of inclusion-exclusion to subtract the non-onto functions from the total number of functions.

Option C (48) is correct because it calculates the exponent of 3 in the prime factorization of 100! using Legendre's formula.

The correct answer is 1540 because it represents the number of ways to choose positions for the Hindi books among the English books.

Option C (${7^5}$) is correct because each of the 5 letters can be posted in any of the 7 boxes independently.

Option A (66) is correct because the committee must have 3 or 4 gentlemen, and the number of ways to form the committee is ${}^4C_3 \times {}^6C_2 + {}^4C_4 \times {}^6C_1 = 66$.

Option D (63) is correct because it represents the total number of ways a student can answer the questions minus the one way to get all answers correct.

Option A ($9 ! \times 2!$) is correct because it treats the two Chief Ministers as one unit and accounts for their internal arrangement.

Option C (3) is correct because solving the factorial equation leads to n = 3.

Option B (120) is correct because it is the total number of five-digit numbers greater than 40000 and divisible by 5 that can be formed using the given digits without repetition.

Option D ($\frac{13!}{40}$) is correct because the number of ways to arrange the guests is calculated by dividing them into two groups and arranging each group around a round table.

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

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