permutation and combination

2011-02-27 8:28 am

1. A 3-digit number is formed from the digits 0, 2, 3, 4, 5, 7 and 9. Each digit can be used only once.
(a) How many even numbers can be formed?
(b) How many odd numbers that are greater than 400 can be formed?

2, There are 12 new buses. Find the number of ways of dividing them into three groups of 4 for three different bus routes.

3. At a music camp, 11 musicians are assigned to 3 bedrooms A,B and C, which have 2 beds, 4 beds and 5 beds repsectively. In how many ways can the musicians be assigned to the beds?

4. A coin is flipped n times, where n >/= 3
(a)Find the number of ways to obtain
(i) exactly 1 Head
(ii) exactly 2 Heads
(iii) at least 2 Heads
(b) It is given that there are 36 ways to obtain exactly 2 heads.
(i) Find n.
(ii) Hence, or otherwise, find the number of ways of obtaining at least 2 heads.

5. Rainbow ice-cream shop offers 25 different flavours of ice-cream. If Jacky orders a double-scoop cone, find the number of combinations of flavours if there are no restrictions.

6. Find the number of ways to distribute 12 different story books among 4 different children evenly.

7. The license plates of cars consist of 2 upper case letters followed by 4 digits, in which the first digit cannot be '0'. If the letters 'I' and 'O' cannot be used and the digits cannot be all the same, then how many different license plates are possible?

回答 (1)

Nemo

2011-03-02 10:22 pm

✔ 最佳答案

我不敢說必定全對，不過答案應該大致如下：
(1)
a. 6P2 + 5P2*2 = 70
b. Consider the number of ways to form a 3-digit odd number greater than 400:
odd-even-odd: 3*3*3 = 27
odd-odd-odd: 3*3*2 = 18
even-even-odd: 1*2*4 = 8
even-odd-odd: 1*4*3 = 12
Total = 27 + 18 + 8 + 12 = 65

(2) 12C4 * 8C4 * 4C4 = 34650

(3) 11P11 = 39916800

(4a)
i. nC1 = n
ii. nC2
iii. Total number of outcomes - no. of no Heads - no. of exactly 1 Head
= 2^n - 1 - n

(4b)
i. nC2 = 36
n!/[(n-2)!(2!)] = 36
Upon solving gives n = 9 or n = -8 (rejected)
ii. 2^9 - 1 - 9 = 502

(5) It is stated that there are no restrictions on how to order a double-scoop cone, so I assume that the same flavour can be ordered twice.
The number of combinations should be:
25C2 + 25 = 325

(6) 12C3 * 9C3 * 6C3 * 3C3 = 369600

(7) Since the digits can be used repeatedly, but cannot all be the same, e.g. 1111, 2222,..... 9999, 9 numbers will be excluded.
The number of different license plates possible is given by:
24P2 * (9*10^3 - 9) = 4963032

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