True Tears question thread

[TrueTears]

First Question:
There are 10 chairs in a row.
a) in how many ways can three people be seated
a) in how many of these will the 2 end chairs be occupied?
b) In how many of these will the 2 end chairs be empty?

a) i can do. its just 10P3
part b and c im just a bit confused. cud some1 help.

Second Question:
A hand of 5 cards is dealt from a normal pack of 52 cards. Find the probability that the hand will contain the ace of spades given that there is at least one ace.

so Pr(AofS | more than 1A)= Pr(AofS n more than 1A) / Pr (more than 1A)
but how do u work out Pr(AofS n more than 1A)? ive tried some ways but they didnt quite work. Could some1 please explain through each step how they would do this question. Thanks

Third Question:
Of the intergers from 1000 to 9999 how many have at least one digit a 5 or 7?
not too sure how to approach this. i did do some workings but dont think it will help if i posted it up here. So could some1 just show their working on how they would do it. Thanks!

Fourth Question:
All possible 3 digit numbers are formed from the odd digits {1,3,5,7,9}
a) how many such numbers are possible if each digit is used only once.
b) How many of the numbers from part a are larger than 350.

b) im not sure how to do lol.

Many thanks to anyone who can help!

[Mao]

1. b)
c)

2. ace of spades given 1 ace:
ace of spades given 2 aces:
ace of spades given 3 aces:
ace of spades given 4 aces:
at least 1 ace: total - no ace =

now add up the four instances, and divide by the total, gives the probability

3.
at least one 5 or 7 = total - (no 5 AND no 7) =

4. a)
b) breaking it down into two parts: 350~399, 400+


PS: please excuse the succinctness, I am not awake enough to explain myself

[TrueTears]

Ahh thanks i get all of it now. but just for 1.b) why is it x 9. In the supplement solutions it said 3P2 x 8P1. where did the 8P1 come from?

[Mao]

oh yes, 8, of course, silly me.

basically, you have three people waiting to be allocated into two end chairs (3P2), and the person who missed out can pick any of the 8 remaining chairs (8P1)

[TrueTears]

just 1 more Q
There are 5 vowels and 21 consonants in the english alphabet. How many different 4 letter 'words' can be formed that contains 2 different vowels and 2 different consonants?

what i did was 5P1 x 4P1 x 21P1 x 20P1= 8400 but the answer is 11025. how do u get that?

[Flaming_Arrow]

i think your way is wrong because it saying its going to be VVCC it could VCVC or CVCV etc.


this is what i got

Pick 2 Vowels and 2 Consonants Then place then in 4P4




however my answer is different from the book

[excal]

i think your way is wrong because it saying its going to be VVCC it could VCVC or CVCV etc.


this is what i got

Pick 2 Vowels and 2 Consonants Then place then in 4P4




however my answer is different from the book

You're sampling without replacement.

Try reducing the sample space for each 'trial'.

I'm way too tired to pump out the equations (and the fact that permutation and combinatoric syntax is way deep in my high level thought processes which I'm not currently capable of).

[shinny]

You're sampling without replacement.

Try reducing the sample space for each 'trial'.

I'm way too tired to pump out the equations (and the fact that permutation and combinatoric syntax is way deep in my high level thought processes which I'm not currently capable of).

I don't think that's the problem...if you account for replacement, that should only serve to INCREASE the number of possible words. chath's answer was 50400, which is more than the book's 11025. I got the same answer as Chath through a different method so it MIGHT be correct. Only thing that worries me is it says 'DIFFERENT' words, but the way me and chath did it should avoid that anyhow...

[excal]

Ok, fine. I'll try and work it out.

[excal]

From a pool of 21 consonants, select 2 letters =

From a pool of 5 vowels, select 2 letters =

The 4 letters can be arranged in ways.

Therefore, the total number of combinations is:



I think the book is wrong.
(apologies to chath for incorrectly pointing out a non-existant mistake!)

[Flaming_Arrow]

xD

[TrueTears]

yeah i got 50400 as well. so did mao, i think the book is wrong.

[Mao]

11025 = 5 * 5 * 21 * 21

meaning, WITH replacement WITHOUT order... which is just stupid...

[TrueTears]

yeah especially when it said u cant have 2 same vowels.... so it has to be without replacing. Book is definitely wrong.

[TrueTears]

Oh and just for this Q.
Morse code consists of a succession of dots and dashes. the symbols formed by the code may consist of one two three or four dots or dashes. How many different symbols may be represented by this code?
how do u do this?