List of top Mathematics Questions on fundamental principle of counting

There are 3 candidate for a Mathematics, 5 for chemistry and 4 for a Physics scholarship. In how many ways can the scholarship be awarded.
There are two main entrances to a building with five floors. Each entrance leads to three lifts and each lift can stop at all the five floors. A person enters the building and reaches a floor. The number of possible ways that the person can reach the floor, is
The total number of 4-digit numbers whose greatest common divisor with 54 is 2 , is ____.
The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is _______.
The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiple of 6, is
Consider 4 boxes, where each box contains 3 red balls and 2 blue balls. Assume that all 20 balls are distinct. In how many different ways can 10 balls be chosen from these 4 boxes so that from each box at least one red ball and one blue ball are chosen?
The total number of 7 digit positive integral numbers with distinct digits that can be formed using the digits 4, 3, 7, 2, 1, 0, 5 is:
The number of 5-digit numbers (no digit is repeated) that can be formed by using the digits \(0,1,2,\ldots,7\) is
How many four digit numbers \( abcd \) exist such that \( a \) is odd, \( b \) is divisible by 3, \( c \) is even and \( d \) is prime?
How many four digit numbers \( abcd \) exist such that \( a \) is odd, \( b \) is divisible by 3, \( c \) is even and \( d \) is prime?
How many four digit numbers \( abcd \) exist such that \( a \) is odd, \( b \) is divisible by 3, \( c \) is even and \( d \) is prime?
A locker in bank has 3 digit lock. Mahesh forgot his password and was trying all possible combinations. He took 6 seconds for each try. The problem was that each digit can be from 0 to 9. How much time will be needed by Mahesh to try all the combinations?
If \(\frac{1}{6!}+\frac{1}{7!}=\frac{x}{8!}\), find x.
Evaluate \(\frac{n!}{\left(n-r\right)!}\), when 
(i) n = 6, r = 2 (ii) n = 9, r = 5