Permutations and Combinations

$10$ distinct points are taken on a circle. Then using these pointsStatement I: The number of triangles that can be formed is $100$Statement II: The number of chords that can be formed is $45$Which of the following is correct?

How many ways can you arrange all the letters and numbers in "KCET 2025" which start with K and end with $5$?

\(\text { The number of diagonals that can be drawn in an octagon is }\)

The number of four digit even number that can be formed using the digits $0,1,2$ and 3 without repetition is

Ten chairs are numbered as 1 to 10. Three women and two men wish to occupy one chair each. First the women choose the chairs marked 1 to 6 , then the men choose the chairs from the remaining. The number of possible ways is

If all permutations of the letters of the word MASK are arranged in the order as in dictionary with or without meaning, which one of the following is 19th word?

A student has to answer 10 questions, choosing at least 4 from each of the parts \(A\) and \(B\). If there are 6 questions in part \(A\) and 7 in part \(B\), then the number of ways can the student choose 10 questions is

\(A\) and \(B\) are non-singleton sets and \(n(A \times B)=35\). If \(B \subset A\), then \({ }^{n(A)} C_{n(B)}\) is equal to

The value of \({ }^{16} C_9+{ }^{16} C_{10}-{ }^{16} C_6-{ }^{16} C_7\) is

If \(P(n): 2^n< n !\) Then the smallest positive integer for which \(P(n)\) is true, is

The number of 4 digit numbers without repetition that can be formed using the digits \(1,2,3,4,5,6,7\) in which each number has two odd digits and two even digits is

Everybody in a room shakes hands with everybody else. The total number of handshakes is 45 . The total number of persons in the room is

The number of ways in which 5 girls and 3 boys can be seated in a row so that no two boys are together is

If ${ }^n C_{12}={ }^n C_8$, then $n$ is equal to