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Let \( S=\{1,2,3,4,5,6,7\} \). Then the number of possible functions \( f:S\to S \) such that \( f(m\cdot n)=f(m)\cdot f(n) \) for every \( m,n\in S \) and \( m\cdot n\in S \) is equal to \( \underline{\quad} \).

All the arrangements, with or without meaning, of the word FARMER are written excluding any word that has two R appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word FARMER in this list is:

Let \( n \) a non-negative integer. Then the number of divisors of the form “\( 4n+1 \)” of the number \( 10^{(10)}\cdot (11)^{11}\cdot (13)^{13} \) is equal to ____________.

The number of six letter words (with or without meaning), formed using all the letters of the word ‘VOWELS’, so that all the consonants never come together, is __________ .

Consider a rectangle \( ABCD \) having \( 5,\,7,\,6,\,9 \) points in the interior of the line segments \( AB,\, CD,\, BC,\, DA \) respectively. Let \( \alpha \) be the number of triangles having these points from different sides as vertices and \( \beta \) be the number of quadrilaterals having these points from different sides as vertices. Then \(( \beta - \alpha )\) is equal to:

Let \( P_{1}, P_{2}, \ldots , P_{15} \) be \( 15 \) points on a circle. The number of distinct triangles formed by points \( P_{i}, P_{j}, P_{k} \) such that \( i + j + k \neq 15 \), is:

There are \( 15 \) players in a cricket team, out of which \( 6 \) are bowlers, \( 7 \) are batsmen and \( 2 \) are wicketkeepers. The number of ways, a team of \( 11 \) players be selected from them so as to include at least \( 4 \) bowlers, \( 5 \) batsmen and \( 1 \) wicketkeeper, is _______.

A number is called a palindrome if it reads the same backward as well as forward. For example \( 285582 \) is a six digit palindrome. The number of six digit palindromes, which are divisible by \( 55 \), is ________.

The sum of all the \( 4 \)-digit distinct numbers that can be formed with the digits \( 1 \), \( 2 \), \( 2 \) and \( 3 \) is:

If \( {}^{n}P_{r} = {}^{n}P_{r+1} \) and \( {}^{n}C_{r} = {}^{n}C_{r-1} \), then the value of \( r \) is equal to:

The number of \( 4 \)-digit numbers which are neither multiple of \( 7 \) nor multiple of \( 3 \) is ______.

If the digits are not allowed to repeat in any number formed by using the digits \( 0, 2, 4, 6, 8 \), then the number of all numbers greater than \( 10{,}000 \) is equal to _______.

Team \( 'A' \) consists of \( 7 \) boys and \( n \) girls and Team \( 'B' \) has \( 4 \) boys and \( 6 \) girls. If a total of \( 52 \) single matches can be arranged between these two teams when a boy plays against a boy and a girl plays against a girl, then \( n \) is equal to:

There are \( 5 \) students in class \( 10 \), \( 6 \) students in class \( 11 \) and \( 8 \) students in class \( 12 \). If the number of ways, in which \( 10 \) students can be selected from them so as to include at least \( 2 \) students from each class and at most \( 5 \) students from the total \( 11 \) students of class \( 10 \) and \( 11 \) is \( 100k \), then \( k \) is equal to ____________ .

The students \( s_1, s_2, \ldots, s_{10} \) are to be divided into \( 3 \) groups \( A \), \( B \) and \( C \) such that each group has at least one student and the group \( C \) has at most \( 3 \) students. Then the total number of possibilities of forming such groups is _______.

If the series \( AB \), \( BC \) and \( CA \) of a triangle \( ABC \) have \( 3 \), \( 5 \) and \( 6 \) interior points respectively, then the total number of triangles that can be constructed using these as vertices, is equal to:

The number of times the digit \( 3 \) will be written when listing the integers from \( 1 \) to \( 1000 \) is ____________.

The number of three-digit even numbers, formed by the digits \( 0 \), \( 1 \), \( 3 \),\( 4 \), \( 6 \), \( 7 \) if the repetition of digits is not allowed, is _________.