CAT Modern Math > Permutation & Combination PYQs

A cafeteria offers 5 types of sandwiches. Moreover, for each type of sandwich, a customer can choose one of 4 breads and opt for either small or large sized sandwich. Optionally, the customer may also add up to 2 out of 6 available sauces. The number of different ways in which an order can be placed for a sandwich, is

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

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

The number of all natural numbers up to $1000$ with non-repeating digits 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 integers greater than $2000$ that can be formed with the digits $0, 1, 2, 3, 4, 5$, using each digit at most once, 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

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

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

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

How many integers in the set $(100,101,102, \ldots, 999)$ have at least one digit repeated?

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 $1,2,3,4,5,6,7,8,9,$ so that in every such number, each digit is used at most once and the digits appear in the ascending order?

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?

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?

Let $AB, CD, EF, GH,$ and $JK$ be five diameters of a circle with center at $O$. In how many ways can three points be chosen out of $A, B, C, D, E, F, G, H, J, K,$ and $O$ so as to form a triangle?

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?