Special numbers and symmetry

Questions 1 to 5 of 8

1. 1. Five people are split into two non-empty teams, and the teams have no names. An AI tool answers C(5, 2) = 10. What is the correct count, and why?

   10, because choosing 2 people from 5 always gives the answer15, because the count is S(5, 2), and the two unnamed teams are not ordered20, because each team can be given a name32, because each person picks one of two teams
2. 2. What is the Bell number B(5), the number of ways to split 5 distinct people into any number of non-empty, unnamed teams?

   Type a number.

3. 3. Match each statement with the family of numbers it describes.

   Choose the matching item for each row.

   Balanced brackets with 3 pairsChoose…Unsigned Stirling number c(4, 2) = 11Bell number B(4) = 15Stirling number S(4, 2) = 7Catalan number C(3) = 5Split 4 distinct items into 2 unnamed non-empty groupsChoose…Unsigned Stirling number c(4, 2) = 11Bell number B(4) = 15Stirling number S(4, 2) = 7Catalan number C(3) = 5Permutations of 4 items that have exactly 2 cyclesChoose…Unsigned Stirling number c(4, 2) = 11Bell number B(4) = 15Stirling number S(4, 2) = 7Catalan number C(3) = 5Split 4 distinct items into any number of groupsChoose…Unsigned Stirling number c(4, 2) = 11Bell number B(4) = 15Stirling number S(4, 2) = 7Catalan number C(3) = 5
4. 4. Six beads sit on a circle, and a rotation moves each bead by 2 places. How many cycles does this rotation have?

   Type a number.

5. 5. An AI tool gives S(4, 2) = 9 for 'split 4 distinct items into 2 unnamed non-empty groups', using the closed form. What did it do wrong?

   It left out the (-1) raised to (k - j) sign in the sum, so the sum is 18 instead of 14It multiplied by 2! where it should have dividedIt used C(4, 2) = 6 in place of S(4, 2)It counted the empty split as one of the answers