📝 Chapter 2: Permutations

Assessment AS Learning — Practice Quiz
🔄 Not Graded — Unlimited Retakes
Purpose: Self-check your skills with factorials, permutations of distinct/identical objects, restrictions, and circular permutations.
Score: 0 / 12
Topic 2.1 — Factorials
Question 1
Evaluate \( 6! \).


Solution:
\( 6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720 \).
Question 2
Simplify \( \dfrac{8!}{6!} \).


Solution:
\( \dfrac{8!}{6!} = \dfrac{8\cdot 7\cdot 6!}{6!} = 8\cdot 7 = 56 \).
Topic 2.2 — Permutations \( P(n,r) \)
Question 3
In how many ways can the gold, silver, and bronze medals be awarded to 8 finalists?


Solution:
\( P(8,3) = \dfrac{8!}{5!} = 8 \cdot 7 \cdot 6 = 336 \).
Question 4
How many 5-letter "words" (any string of letters) can be formed from a 26-letter alphabet with NO repetition?
Solution:
Order matters, no repetition: \( P(26,5) = 26 \cdot 25 \cdot 24 \cdot 23 \cdot 22 = 7\,893\,600 \).
Topic 2.3 — Permutations with Identical Objects
Question 5
How many distinct arrangements of the letters of "BOOK" are possible?


Solution:
4 letters with O repeated twice: \( \dfrac{4!}{2!} = \dfrac{24}{2} = 12 \).
Question 6
How many distinct arrangements of the letters of "MISSISSIPPI"?


Solution:
11 letters: 1 M, 4 I, 4 S, 2 P. \( \dfrac{11!}{1!\, 4!\, 4!\, 2!} = \dfrac{39\,916\,800}{1\cdot 24\cdot 24\cdot 2} = 34\,650 \).
Topic 2.4 — Permutations with Restrictions
Question 7
Five people sit in a row. In how many ways can they sit if Alice and Bob must sit together?


Solution:
"Glue" Alice and Bob: 4 units arrange in \( 4! = 24 \) ways. Internal AB or BA: \( 2! = 2 \). Total: \( 24 \cdot 2 = 48 \).
Question 8
How many 4-letter "words" can be made from letters A, B, C, D, E if the word must START with a vowel (A or E) and have no repetition?


Solution:
First letter: 2 vowels. Remaining 3 letters from 4: \( P(4,3) = 24 \). Total: \( 2 \cdot 24 = 48 \).
Topic 2.5 — Circular Permutations
Question 9
How many ways can 6 distinct people sit around a circular table?


Solution:
Circular permutation: \( (n-1)! = 5! = 120 \) (one seat is fixed by rotation symmetry).
Question 10
In how many ways can 8 distinct beads be arranged on a bracelet (which can be flipped)?
Solution:
Bracelet = circle with reflection. \( \dfrac{(n-1)!}{2} = \dfrac{7!}{2} = 2520 \).
Mixed Application
Question 11
Solve for \( n \) in \( P(n,2) = 30 \). (\( n \in \mathbb{N} \))


Solution:
\( n(n-1) = 30 \Rightarrow n^2 - n - 30 = 0 \Rightarrow (n-6)(n+5)=0 \Rightarrow n = 6 \).
Question 12
How many 4-digit numbers can be formed from {1,2,3,4,5,6,7} without repetition such that the number is even?


Solution:
Last digit even: 3 choices (2, 4, 6). Then 3 remaining positions from 6 remaining digits: \( P(6,3) = 120 \). Total: \( 3 \cdot 120 = 360 \).