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Arranging n different objects yields n! orders, and the permutations of r objects from n obey nPr = n! ÷ (n − r)!, so this IB counting sheet rests on factorials and arrangements. The key also records nCr, 120 arrangements, and 6 permutations.
Key Takeaways
The number of arrangements of n different objects is n! (Q1).
nPr = n! ÷ (n − r)! (Q2).
The key records nCr and an answer of 120 arrangements.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.
The number of ways to arrange n different objects is
(a) n! (b) 2ⁿ (c) n² (d) n
2.
The number of permutations of r objects chosen from n is given by
(a) nPr = n! ÷ (n − r)! (b) nPr = n! ÷ r! (c) nPr = n! × r! (d) nPr = (n − r)!
3.
The number of combinations of r objects chosen from n is given by
(a) nCr = n! ÷ (r! × (n − r)!) (b) nCr = n! ÷ r! (c) nCr = (n − r)! (d) nCr = n! × r!
4.
In a combination, the order of selection
(a) does not matter (b) always matters (c) is always fixed (d) only matters for large n
5.
The value of 5! is
(a) 120 (b) 25 (c) 15 (d) 100
6.
The number of ways to arrange the letters A, B and C is
(a) 6 (b) 3 (c) 9 (d) 1
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.
Evaluate 4!.
8.
Distinguish between a permutation and a combination.
9.
Write the formula for nPr.
10.
Write the formula for nCr.
11.
In how many ways can the letters of the word CAT be arranged?
12.
State whether choosing a team of 3 players from 10 players is a permutation or a combination.
13.
In how many ways can 3 different books be arranged on a shelf?
14.
What is the value of 0!?
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.
How many ways can the letters of the word MATH be arranged?
16.
A code is made from 4 different digits chosen from 0 to 9. Using nPr with n = 10 and r = 4, find the number of possible codes.
17.
How many ways can a committee of 3 be chosen from 8 students? Use the combination formula 8C3.
18.
In how many ways can 5 different books be arranged on a shelf?
19.
A team of 2 is chosen from 6 players. Using 6C2, find the number of possible teams.
20.
How many ways can you select 2 letters from the 5 letters of the word PLANE? Use 5C2.
Answer Key
1. a) n!
2. a) nPr = n! ÷ (n − r)!
3. a) nCr = n! ÷ (r! × (n − r)!)
4. a) does not matter
5. a) 120
6. a) 6
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
How many ways can n different objects be arranged?
n!, option (a) in this worksheet (Q1).
What is the formula for permutations nPr?
n! ÷ (n − r)!, option (a) in this worksheet (Q2).
How many arrangements are recorded in the key?
120.