1. There are 6 math books and 5 language books on different shelves. The number of ways to choose 1 of each is 6 × 5 = 30. 2. There are 5 colors of tops and 4 colors of skirts. The total number of dress combinations is 5 × 4 = 20. There are 3 styles of shoes, so the total number of styles is 3. 3. The number of 3-digit numbers that can be formed without repeating digits is 100 × 99 × 98 = 9,702. The number of ways for 2 boys to sit in 5 chairs is 5 × 4 = 20.

1. PERMUTATION
& COMBINATION

2. 𝒑𝒈 𝟏𝟏𝟕
6 𝑀𝑎𝑡ℎ𝑠 𝑏𝑜𝑜𝑘 𝑜𝑛 𝑢𝑝𝑝𝑒𝑟 𝑠ℎ𝑒𝑙𝑓. 5 𝑙𝑎𝑛𝑔𝑢𝑎𝑔𝑒 𝑏𝑜𝑜𝑘 𝑜𝑛 𝑙𝑜𝑤𝑒𝑟 𝑠ℎ𝑒𝑙𝑓. 𝑁 (1 𝑀𝑎𝑡ℎ𝑠 & 1 𝐿𝑎𝑛𝑔𝑢𝑎𝑔𝑒) =?
1 𝑀𝑎𝑡ℎ𝑠 × 1 𝐿𝑎𝑛𝑔𝑢𝑎𝑔𝑒

3. 𝒑𝒈 𝟏𝟏𝟕
5 𝑐𝑜𝑙𝑜𝑢𝑟𝑠 𝑜𝑓𝑡𝑜𝑝.
4 𝑐𝑜𝑙𝑜𝑢𝑟𝑠 𝑜𝑓 𝑠𝑘𝑖𝑟𝑡.
𝛴 𝑑𝑟𝑒𝑠𝑠 𝑐𝑜𝑚𝑏𝑖𝑛𝑎𝑡𝑖𝑜𝑛 = ?
3 𝑠𝑡𝑦𝑙𝑒𝑠 𝑜𝑓 𝑠ℎ𝑜𝑒𝑠.
𝛴 𝑠𝑡𝑦𝑙𝑒 = ?

4. 𝒑𝒈 𝟏𝟏𝟖
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 3 𝑑𝑖𝑔𝑖𝑡 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
𝑐𝑎𝑛 𝑏𝑒 𝑓𝑜𝑟𝑚𝑒𝑑
𝑤𝑖𝑡ℎ𝑜𝑢𝑡 𝑟𝑒𝑝𝑒𝑎𝑡𝑖𝑛𝑔?
5 𝑐ℎ𝑎𝑖𝑟𝑠 𝑓𝑜𝑟 2 𝑏𝑜𝑦𝑠 𝑡𝑜 𝑠𝑖𝑡.
𝑁 = ?
1𝑠𝑡 𝑏𝑜𝑦 × 2𝑛𝑑 𝑏𝑜𝑦

6. 𝒑𝒈 𝟏𝟏𝟖
20
42
200
360
120

7. 𝒑𝒈 𝟏𝟏𝟖
18
16
78

8. 𝒑𝒈 𝟏𝟏𝟗
𝐴, 𝐵, 𝐶 𝑖𝑛 𝑎 𝑟𝑜𝑤. 𝛴 𝑎𝑟𝑟𝑎𝑛𝑔𝑒𝑚𝑒𝑛𝑡?
3 2 1

9. 𝒑𝒈 𝟏𝟏𝟗
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 2 𝑑𝑖𝑔𝑖𝑡 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑐𝑎𝑛 𝑏𝑒 𝑓𝑜𝑟𝑚𝑒𝑑 𝒘𝒊𝒕𝒉𝒐𝒖𝒕 𝒓𝒆𝒑𝒆𝒂𝒕𝒊𝒏𝒈 ?

10. 𝒑𝒈 𝟏𝟐𝟐

11. 𝑛𝑃𝑟

12. 𝒑𝒈 𝟏𝟐𝟏
4𝑃2
=
4 !
(4 − 2) !
=
4 × 3 × 2 !
2 !
= 4 × 3
7𝑃3
=
7 !
(7 − 3) !
=
7 × 6 × 5 × 4 !
4 !
= 7 × 6 × 5
12𝑃4
=
12 !
(12 − 4) !
=
12 × 11 × 10 × 9 × 8 !
8 !
= 12 × 11 × 10 × 9

13. 𝒑𝒈 𝟏𝟐𝟒
𝑎𝑏, 𝑎𝑐, 𝑏𝑎, 𝑏𝑐, 𝑐𝑎, 𝑐𝑏
𝑎𝑏𝑐, 𝑎𝑏𝑑, 𝑎𝑐𝑑, 𝑎𝑐𝑏, 𝑎𝑏𝑑, 𝑎𝑑𝑐,
𝑏𝑎𝑐, 𝑏𝑎𝑑, 𝑏𝑐𝑑, 𝑏𝑐𝑎, 𝑏𝑑𝑎, 𝑏𝑑𝑐,
𝑐𝑎𝑏, 𝑐𝑎𝑑, 𝑐𝑏𝑑, 𝑐𝑏𝑎, 𝑐𝑑𝑎, 𝑐𝑑𝑏,
𝑑𝑎𝑏, 𝑑𝑎𝑐, 𝑑𝑏𝑐, 𝑑𝑏𝑎, 𝑑𝑐𝑎, 𝑑𝑐𝑏.
32 760 970 200
1 568
5 040 336
21
𝑛2
+ 11𝑛 + 30 𝑛2
+ 𝑛 𝑛2
+ (1 − 2𝑟) 𝑛 + 𝑟 (𝑟 − 1)

14. 𝒑𝒈 𝟏𝟐𝟑
𝑡𝑎𝑘𝑒 3 𝑙𝑒𝑡𝑡𝑒𝑟𝑠 𝑓𝑟𝑜𝑚 “𝑣𝑜𝑙𝑢𝑚𝑒”
4 𝑣𝑜𝑤𝑒𝑙𝑠, 4 𝑐𝑜𝑛𝑠𝑜𝑛𝑎𝑛𝑡𝑠 𝑖𝑛 𝑎 𝑟𝑜𝑤. 𝑇𝑜𝑡𝑎𝑙 8 𝑙𝑒𝑡𝑡𝑒𝑟𝑠. 𝛴 𝑣𝑜𝑤𝑒𝑙 𝑎𝑡 𝑏𝑜𝑡ℎ 𝑒𝑛𝑑𝑠?
𝑣𝑜𝑤𝑒𝑙𝑠 × 𝑐𝑜𝑛𝑠𝑜𝑛𝑎𝑛𝑡𝑠

15. 𝒑𝒈 𝟏𝟐𝟑
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 4 𝑑𝑖𝑔𝑖𝑡 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑐𝑎𝑛 𝑏𝑒 𝑓𝑜𝑟𝑚𝑒𝑑 ?
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 4 𝑑𝑖𝑔𝑖𝑡 𝒐𝒅𝒅 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑐𝑎𝑛 𝑏𝑒 𝑓𝑜𝑟𝑚𝑒𝑑 ?
4𝑃1

17. 𝒑𝒈 𝟏𝟐𝟒

18. 𝒑𝒈 𝟏𝟐𝟒

19. 𝒑𝒈 𝟏𝟐𝟒

20. 𝒑𝒈 𝟏𝟐𝟔
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑛𝑜𝑛 − 𝑒𝑝𝑒𝑎𝑡𝑖𝑛𝑔
𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑐𝑎𝑛 𝑏𝑒 𝑓𝑜𝑟𝑚𝑒𝑑?
𝐶𝑎𝑠𝑒 1 ∶ 1 𝑑𝑖𝑔𝑖𝑡
𝐶𝑎𝑠𝑒 2 ∶ 2 𝑑𝑖𝑔𝑖𝑡
𝐶𝑎𝑠𝑒 3
∶ 3 𝑑𝑖𝑔𝑖𝑡

21. 𝒑𝒈 𝟏𝟐𝟔
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 3 𝑑𝑖𝑔𝑖𝑡 𝑛𝑜𝑠
𝑐𝑎𝑛 𝑏𝑒 𝑓𝑜𝑟𝑚𝑒𝑑
(𝑛𝑜𝑛 − 𝑟𝑒𝑝𝑒𝑎𝑡𝑖𝑛𝑔) ?

24. 𝒑𝒈 𝟏𝟐𝟕
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑜𝑑𝑑 𝑛𝑜𝑠 (𝑛𝑜𝑛 − 𝑟𝑒𝑝𝑒𝑎𝑡𝑖𝑛𝑔) ?
𝐶𝑎𝑠𝑒 1 ∶
5, 7 𝑡ℎ𝑜𝑢𝑠𝑎𝑛𝑑
4
𝐶𝑎𝑠𝑒 2 ∶
𝟒, 6, 8 𝑡ℎ𝑜𝑢𝑠𝑎𝑛𝑑 4
3𝑃1
3𝑃1
840
840 1288

25. 𝒑𝒈 𝟏𝟐𝟖
8 𝑝𝑙𝑎𝑦𝑒𝑟𝑠 𝑖𝑛 𝑎 𝑟𝑜𝑤.
If someone if designated
not to stand
in the 1st & last posit𝑖𝑜𝑛.
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑤𝑎𝑦𝑠?
1𝑠𝑡 𝑝𝑒𝑟𝑠𝑜𝑛 × 7 𝑝𝑒𝑟𝑠𝑜𝑛𝑠
1𝑠𝑡 𝑝𝑒𝑟𝑠𝑜𝑛 (1𝑠𝑡 & 𝑙𝑎𝑠𝑡 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛) × 7 𝑝𝑒𝑟𝑠𝑜𝑛𝑠

26. 𝒑𝒈 𝟏𝟐𝟖
5𝑀, 5𝐹 𝑖𝑛 𝑎 𝑟𝑜𝑤.
(𝑎) 2F next to each other
(b) 2F not next to each other