A list of Counting method & Probability questions:::: (1 Viewer)

[coeyz]

Please help
And I'm still confuse about permutation and combination.
permutation - order is important. its about arrangement
combination- order is not important. its about selection
But if the question doesnt contain those kind of words like "arrange".. "choose"... how do we know whether the order is important ? how can we distinguish which is permutation/combination?

THANKS A LOT!


1) How many odd numbers, greater than 500,000 can be made from the digits 2,3,4,5,6,7 without repetition? (168)

I did it like this: why is this wrong???
3 x 4! x 3

2) Three letters from the word RELATION are arranged in a row. In how many was can this be done?
How many of these contain exactly one vowel? (336, 144)

First one I did it, but the second one I did like 4P1 x 4P2 x 3! and couldnt get the answer

3) A man stays 3days at a hotel and the menu is the same for breakfast each day. He may have one of three type of egg dish, or two types of fish, or meat. In how many ways can he order his 3breakfasts if he does not have eggs two
days running nor does he repeat any dish? (78)

4) A boy has 5 blue marbles, 4 green marbles and 3 red marbles. In how many ways can he arrange 4 of them in a row, if the marbles of one colour are indistinguishable? (80)

5) Five letters the word DRILLING are arranged in a row. Find the number of ways in which this is to be done, when the first letter is I and the last letter is L.
a) If no letter may be repeated (24)
b) If each letter may occur as many times as it does in DRILLING. (120)

6) A man, who works a 5day week, can go to work on foot, by cycle or by bus. How many ways can he arrange a week's
travel to work? (243)

7) In a class of 30 pupils, one prize is awarded for English, another for French and 3rd for Math. In how many ways can the recipients be chosen? (127000)

8) The computer department in a large company assigns a personal code number to each employee in the form of a 3-digit number, using 0,1,2...9 inclusive. Code numbers starting with zero are reserved for members of
the management. How many code numbers are reserved for non-management employees? (900)

9) There are 16 books on a shelf. In how many ways can these be arranged if 12 of them are of history and must be kept in a given order? (120)

 

[khorne]

Please help
And I'm still confuse about permutation and combination.
permutation - order is important. its about arrangement
combination- order is not important. its about selection
But if the question doesnt contain those kind of words like "arrange".. "choose"... how do we know whether the order is important ? how can we distinguish which is permutation/combination?

THANKS A LOT!


1) How many odd numbers, greater than 500,000 can be made from the digits 2,3,4,5,6,7 without repetition? (168)

I did it like this: why is this wrong???
3 x 4! x 3

2) Three letters from the word RELATION are arranged in a row. In how many was can this be done?
How many of these contain exactly one vowel? (336, 144)

First one I did it, but the second one I did like 4P1 x 4P2 x 3! and couldnt get the answer

3) A man stays 3days at a hotel and the menu is the same for breakfast each day. He may have one of three type of egg dish, or two types of fish, or meat. In how many ways can he order his 3breakfasts if he does not have eggs two
days running nor does he repeat any dish? (78)

4) A boy has 5 blue marbles, 4 green marbles and 3 red marbles. In how many ways can he arrange 4 of them in a row, if the marbles of one colour are indistinguishable? (80)

5) Five letters the word DRILLING are arranged in a row. Find the number of ways in which this is to be done, when the first letter is I and the last letter is L.
a) If no letter may be repeated (24)
b) If each letter may occur as many times as it does in DRILLING. (120)

6) A man, who works a 5day week, can go to work on foot, by cycle or by bus. How many ways can he arrange a week's
travel to work? (243)

7) In a class of 30 pupils, one prize is awarded for English, another for French and 3rd for Math. In how many ways can the recipients be chosen? (127000)

8) The computer department in a large company assigns a personal code number to each employee in the form of a 3-digit number, using 0,1,2...9 inclusive. Code numbers starting with zero are reserved for members of
the management. How many code numbers are reserved for non-management employees? (900)

9) There are 16 books on a shelf. In how many ways can these be arranged if 12 of them are of history and must be kept in a given order? (120)

1) consider two cases:
odd number first
2x4!x2 = 96
and 6 first:
1x4!x3 = 72

Add them = 168

2)
Consider that you choose one vowel...if first there are 12 permutations, 2nd there are 12 and third there are 12. So 36 permutations with one vowel. If you multiply it by 4 (4 vowels) (4C1) you get 144.

I'll leave the rest...a bit lazy

But for 3) try finding the total possible combinations and then eliminating the ones which don't fit.

6) = 3x3x3x3x3

8) total codes = 10x10x10 = 1000
codes starting with 0 = 1x10x10 = 100
therefore remaining codes = 900


I'm sorry, but it doesn't appear that you have tried on ANY of these...Half of them are dead set easy

 

[coeyz]

1) consider two cases:
odd number first
2x4!x2 = 96
and 6 first:
1x4!x3 = 72

Add them = 168

2)
Consider that you choose one vowel...if first there are 12 permutations, 2nd there are 12 and third there are 12. So 36 permutations with one vowel. If you multiply it by 4 (4 vowels) (4C1) you get 144.

I'll leave the rest...a bit lazy

But for 3) try finding the total possible combinations and then eliminating the ones which don't fit.

6) = 3x3x3x3x3

8) total codes = 10x10x10 = 1000
codes starting with 0 = 1x10x10 = 100
therefore remaining codes = 900


I'm sorry, but it doesn't appear that you have tried on ANY of these...Half of them are dead set easy

NO I have tried them all.
I wont post it here bofore trying
anyway thanks I will try it again

 

[z3288301]

Here are my answers for the one's Khorne missed:

3) There are 3 egg meals, 2 fish meals and 1 meat meal which gives 6 meals altogether. Without restrictions this can be arranged in 6x5x4=120 ways. We now subtract subtract the number of ways which involves 2 egg meals in a row.

No. of ways in which the first 2 meals are egg: 3x2x4=24
No. of ways in which the last 2 meals are egg: 4x3x2=24
Adding these we have 48.
However, there is overcounting here as both cases would include the situations that have 3 meals of egg. therefore we subtract 3x2x1=6 giving 42 ways.

Thus, 120-42=78

4) There are 5 blue marbles, 4 green marbles and 3 red marbles. We will first treat the question as if there were 4 blue, 4 green and 4 red. This gives 3x3x3x3=81 ways. However, there are actually only 3 red marbles. Thus you cannot have the arrangement red red red red so you must deduct 1. This gives 80.

5)Since the first letter must be i and the last letter must be l then the problem can be considered as the number of ways of forming a 3 letter word rather than the number of ways of forming a 5 letter word.

a. Since i,l cannot be used again we have, 4x3x2=24
b. Since i,l can be reused we have, 6x5x4=120

 

[z3288301]

9) since the 12 books have to be together in a given order you treat them as 1 unit. So this problem is the same as arranging 4+1=5 books giving 5!=120.

 

[coeyz]

Thanks guyss ~~

 

[coeyz]

Here are my answers for the one's Khorne missed:

3) There are 3 egg meals, 2 fish meals and 1 meat meal which gives 6 meals altogether. Without restrictions this can be arranged in 6x5x4=120 ways. We now subtract subtract the number of ways which involves 2 egg meals in a row.

No. of ways in which the first 2 meals are egg: 3x2x4=24
No. of ways in which the last 2 meals are egg: 4x3x2=24
Adding these we have 48.
However, there is overcounting here as both cases would include the situations that have 3 meals of egg. therefore we subtract 3x2x1=6 giving 42 ways.

Thus, 120-42=78

4) There are 5 blue marbles, 4 green marbles and 3 red marbles. We will first treat the question as if there were 4 blue, 4 green and 4 red. This gives 3x3x3x3=81 ways. However, there are actually only 3 red marbles. Thus you cannot have the arrangement red red red red so you must deduct 1. This gives 80.

I dont really get wht you said for question 4..
dont we just pick 4 balls from 12 different balls and arrange it?

 

[z3288301]

First treat the question as if there were 4 blue, 4 green and 4 red marbles. So the first marble you pick can be blue, green or red giving 3 choices. Similarly the 2nd, 3rd and 4th also have 3 choices giving a total of 3x3x3x3=81 combinations. However there are actually only 3 red marbles thus you cannot have red red red red so you must subtract one to give 80. The fact that there are 5 blue marbles does not affect the counting since we only need 4.

 

[omniscience]

u don't know how to count?