Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is ____.
Let S = {1, 2, 3, ..., 9}. For k = 1, 2, ..., 5, let Nk be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N1 + N2 + N3 + N4 + N5 = ?
A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then, the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is
In an exciting finish to an one-day cricket match between India and Pakistan, the Indians require 10 runs in the last 3 balls to win. If any one of the scores 0, 1, 2, 3, 4, 6 can be made from a ball and no wides or no-balls are bowled then in how many different sequences can the batsmen make exactly 10 runs?
The number of strictly increasing functions $f$ from the set $\{1,2,3,4,5,6\}$ to the set $\{1,2,3,\ldots,9\}$ such that $f(i)\neq i$ for $1\leq i\leq6$, is equal to:
Let $S=\{1,2,3,4,5,6,7,8,9\}$. Let $x$ be the number of 9-digit numbers formed using the digits of $S$ such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9-digit numbers formed using the digits of $S$ such that only two digits are repeated and each of these is repeated exactly twice. Then
Given that \(A = \{x_1, x_2, x_3, \ldots, x_7\}\); \(B = \{y_1, y_2, y_3\}\). 3 elements in \(A\) having image \(y_2\) can be chosen in \(^7C_3\) ways. Now we are left with 4 elements in \(A\) which are to be associated with \(y_1\) or \(y_3\), that is, each of 4 elements \(A\) has 2 choices \(y_1\) or \(y_3\), that is, in \((2)^4\) ways. But there are 2 ways when one element of \(B\) will remain associated. The required number of onto functions from \(A\) to \(B\) such that exactly 3 elements of \(A\) map to \(y_2\) is: