Permutations and Combinations — Class 12 Mathematics PYQs

What is the sum of all four digit numbers formed by using all digits $0,1,4,5$ without repetition of digits?

A man has $7$ relatives ($4$ women and $3$ men). His wife also has $7$ relatives ($3$ women and $4$ men). In how many ways can they invite $3$ women and $3$ men so that $3$ of them are man's relatives and $3$ of them are his wife's relatives?

A triangle $PQR$ is such that $3$ points lie on the side $PQ$, $4$ points on $QR$ and $5$ points on $RP$ respectively. Triangles are constructed using these points as vertices. What is the number of triangles so formed?

If $26!=n{8}^k$, where $k$ and $n$ are positive integers, then what is the maximum value of $k$?

How many four-digit natural numbers are there such that all of the digits are even?

Four digit numbers are formed by using the digits $1,2,3,5$ without repetition of digits. How many of them are divisible by $4$?

What is the number of $6$-digit numbers that can be formed only by using $0,1,2,3,4$ and $5$ (each once); and divisible by $6$?

What is the maximum number of points of intersection of $10$ circles ?

Consider the following statements for a fixed natural number $n$ :

1. $C(n,r)$ is greatest if $n=2r$
2. $C(n,r)$ is greatest if $n=2r-1$ and $n=2r+1$

Which of the statements given above is/are correct?

$m$ parallel lines cut $n$ parallel lines giving rise to $60$ parallelograms. What is the value of $(m+n)$?