Permutations II: restrictions
As a refresher, try the last question from Permutations I.
1. A 9-person race does not allow ties. Alice, Bob, and Carol are all in the race. In how many different orders can the results be, if Alice placed somewhere before Carol and Carol placed somewhere before Bob?
2. Using the distinct digits 0 to 9 at most once, how many 4-digit numbers have the first digit (ie. thousands place) being the smallest digit?
Hint 1
Consider the combinations of the 4 numbers, and the number of permutations available per combination.
3. How many 5-digit numbers, containing only different digits from 1 to 9, have exactly two even digits?
Hint 1
The decisions are 1) choosing 2 even digits, 2) choosing 3 odd digits, and 3) permuting the 5 digits.
4. How many 5-digit numbers, containing only different digits from 0 to 9, have exactly two even digits?
5. A teacher of 11 students wants to keep 4 chatty students apart in a line, so that none of the 4 students are next to each other. How many ways to line up the 11 students?
Hint 1
Each chatty student could be next to a non-chatty at the beginning or end of the line, or they could be between two non-chatty.
Hint 2
Assume there is some order of non-chatty students, and then find out how and where to add the chatty students.
6. A teacher of 11 students wants to keep A not next to B, and at the same time, C not next to D. How many ways to line up the 11 students?
Hint 1
Express using the ways of 1) any line up 2) A next to B 3) C next to D and 4) A next to B, and also C next to D. Draw a Venn diagram if it helps.
Hint 2
Recall the last question from Permutations I.
7. The following diagram has nine non-overlapping triangles. Triangles a and b share a side; triangles a and c do not. How many ways are there to assign the labels 1, 2, 3, 4 to four different triangles, so that no two triangles share a side?
Hint 1
Consider the 4 triangles have 0, 1, 2, or 3 of the downward-pointing triangles in the middle.
