The sum of all four-digit numbers that can be formed with the distinct non-zero digits $a, b, c$ , and $d$ , with each digit appearing exactly once in every number, is $153310+n$ , where $n$ is a single digit natural number. Then, the value of $(a+b+c+d+n)$ is

P, Q, R and S are four towns. One can travel between P and Q along 3 direct paths, between Q and S along 4 direct paths, and between P and R along 4 direct paths. There is no direct path between P and S, while there are few direct paths between Q and R, and between R and S. One can travel from P to S either via Q, or via R, or via Q followed by R, respectively, in exactly 62 possible ways. One can also travel from Q to R either directly, or via P, or via S, in exactly 27 possible ways. Then, the number of direct paths between Q and R is

What is the number of all positive integers less than or equal to 500 that have non-repeating digits?

The number of all positive integers up to 500 with non-repeating digits is

The number of all natural numbers up to 1000 with non-repeating digits is

The number of ways of distributing 20 identical balloons among 4 children such that each child gets some balloons but no child gets an odd number of balloons, is

The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1421, including itself, is

The number of groups of three or more distinct numbers that can be chosen from 1, 2, 3, 4, 5, 6, 7 and 8 so that the groups always include 3 and 5, while 7 and 8 are never included together is

The number of ways of distributing 15 identical balloons, 6 identical pencils and 3 identical erasers among 3 children, such that each child gets at least four balloons and one pencil, is

A four-digit number is formed by using only the digits 1, 2 and 3 such that both 2 and 3 appear at least once. The number of all such four-digit numbers is

How many 4-digit numbers, each greater than 1000 and each having all four digits distinct, are there with 7 coming before 3?

With rectangular axes of coordinates, the number of paths from $(1,1)$ to $(8,10)$ via $(4,6)$, where each step from any point $(x,y)$ is either to $(x,y+1)$ or to $(x+1,y)$ is

How many numbers with two or more digits can be formed with the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 so that in every such number, each digit is used at most once and the digits appear in the ascending order? [TITA]

In a tournament, there are 43 junior level and 51 senior level participants. Each pair of juniors play one match. Each pair of seniors play one match. There is no junior versus senior match. The number of girl versus girl matches in junior level is 153, while the number of boy versus boy matches in senior level is 276. The number of matches a boy plays against a girl is

In how many ways can 7 identical erasers be distributed among 4 kids in such a way that each kid gets at least one eraser but nobody gets more than 3 erasers ?

How many four digit numbers, which are divisible by 6, can be formed using the digits 0, 2, 3, 4, 6, such that no digit is used more than once and 0 does not occur in the left-most position? [TITA]

In how many ways can 8 identical pens be distributed among Amal, Bimal, and Kamal so that Amal gets at least 1 pen, Bimal gets at least 2 pens, and Kamal gets at least 3 pens? [TITA]