The total number of \( 5 \)-digit numbers, formed by using the digits \( 1, 2, 3, 5, 6, 7 \) without repetition, which are multiple of \( 6 \), is:
\( 36 \)
\( 48 \)
\( 60 \)
\( 72 \)
The number of ways to distribute \(30\) identical candies among four children \(C_1, C_2, C_3\) and \(C_4\) so that \(C_2\) receives atleast \(4\) and atmost \(7\) candies, \(C_3\) receives atleast \(2\) and atmost \(6\) candies, is equal to:
The number of natural numbers lying between \( 1012 \) and \( 23421 \) that can be formed using the digits \( 2, 3, 4, 5, 6 \) (repetition of digits is not allowed) and divisible by \( 55 \) is ________.
A class contains \( b \) boys and \( g \) girls. If the number of ways of selecting \( 3 \) boys and \( 2 \) girls from the class is \( 168 \), then \( b + 3g \) is equal to ______________.
In an examination, there are \( 5 \) multiple choice questions with \( 3 \) choices, out of which exactly one is correct. There are \( 3 \) marks for each correct answer, \( -2 \) marks for each wrong answer and \( 0 \) mark if the question is not attempted. Then, the number of ways a student appearing in the examination gets \( 5 \) marks is _________.
Let \( S \) be the set of all passwords which are six to eight characters long, where each character is either an alphabet from \( \{A,B,C,D,E\} \) or a number from \( \{1,2,3,4,5\} \) with the repetition of characters allowed. If the number of passwords in \( S \) whose at least one character is a number from \( \{1,2,3,4,5\} \) is \( \alpha \times 5^{6} \), then \( \alpha \) is equal to ________.
The total number of four digits numbers such that each of first three digits is divisible by the last digit, is equal to ______.
The letters of the word ‘\( \text{MANKIND} \)’ are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word ‘\( \text{MANKIND} \)’ is ______.
The number of bijective functions \( f:\{1,3,5,7,\ldots,99\}\to\{2,4,6,8,\ldots,100\} \), such that \( f(3)\ge f(9)\ge f(21)\ge \ldots \ge f(99) \), is ______________.
Numbers are to be formed between \( 1000 \) and \( 3000 \), which are divisible by \( 4 \), using the digits \( 1,2,3,4,5, \text{and}\ 6 \) without repetition of digits. Then the total number of such numbers is ______.
The total number of three-digit numbers, with one digit repeated exactly two times, is __________.
