Counting Techniques

[math magic]

Need help with the following questions. If anyone could help it would be greatly appreciated. Thanks

Determine the probablity that a permutation of the letters of the word MATHS will feature the letter H further to the right than the T.

Ten different books are randomly arranged in a row. Three of the ten are by the same author. Determine the probability that the random arrangement has these three books together.

A security code consists of four different letters chosen from the 26 in the alphabet, order being important.
a) If one such code is created at random what is the probability that it will contain at least one voewl?
b) Given that the code does not contain any vowels what is the probability that it will contain an X

What is the probability that a hand of five cards will contain four of one kind.

A four digit number is to be formed using the digits 2, 3, 4, 5, 6, 7 and 8 with no digit used more than once in anumber. What is the probability that the number formed will be even or more than 7000?

Twelve people are placed in four teams of three:
Team 1: Alex, Jules, Tony
Team 2: Karlee, Pat, Toni
Team 3: Benny, Megan, Tegan
Team 4: Clare, Joh, Harvey
If four of these people are randomly chosen to attend a function what is the probability that the four will
a) include Toni?
b) involve one person from each team?
c) not include anyone from team two?
d) include all three people from team four?

A security code consists of three different letters chosen from the 26 in the alphabet followed by one digit from 1 to 9. If one code is chosen at random what is the probability that it will be one that does include an A?

A child is told she can bring five toys with her one holiday. The child decides to choose the five from 6 jigsaws, 8 dolls, 4 balls, 2 trucks.
How many different sets of five toys are there?
How many of these sets have at least one from each of the four categories of toys?

Five places exist on a course for teachers. The five are to be chosen from the 11 teachers. The 11 come from three districts with 6 from district A, 4 from district B and 1 from district C.
How many different groups of five are there?
How many different groups are there if each group must contain at least one from each three districts?

Security codes are to be allocated with either 2 digits followed by 3 letters or 3 digits followed by 2 letters. Any of the digits from 0 to 9 can be used and any of the letters A to Z can be used but no digit and no letter may feature in a particular code more than once.
How many codes are possible if
a) code must not end with Z?
b) code must start with 1 and end with T?
c) code must contain a 9?
d) code must contain a 5?
e) code must contain at least once of digits 5 and 9?

A six character code is to be formed using four different letters chosen from the six letters A, B, C, D, E and F and two different digits chosen from the four digits 1, 2, 3 and 4. How many different codes are possible if the letters and digits can feature in any order?

Ten candidates for a particular job are to be called for an interview, five on Monday and five on Tuesday. One of the candidates can only attend on the Tuesday but all others have indicated they are available for either day. In how many different orders can the Monday interviews be arranged if the candidates to be interviewed on the Tuesday will be decided after the interviews for Monday are decided?

Ten books are to be arranged on a shelf. Three of the ten books are by one author and the other books are all by different authors. How many arrangements are possible if the three by the same author
a) must be kept together but in no particular order?
b) must be kept together at the left end of the shelf and in a particular order?

The numbers 1, 2, 3, 4, 5 and 6 and the vowels a, e, i, o and u are written one to a card and the eleven cards are then arranged in a line.
a) In how many of the arrangements are all the vowels together in the order a, e, i, o, u?
b) In how many arrangements are all the vowels together?
c) In how many of the arrangements are all the numbers together in the order 1, 2, 3, 4, 5, 6?
d) How many of the arrangements are all the numbers together?

[Rich2189]

perms and combination questions, great laugh these questions.

for you question about books. just do 7! = 5040 for part a). Dunno about the rest, perms and cons are dodgey.

[zaza]

Is this all, or do you have any more homework for us to do...?

[krtxmrtz]

Determine the probablity that a permutation of the letters of the word MATHS will feature the letter H further to the right than the T.

The total number of ppossible permutations is 5! (5 different letters)

Then you have 4 "favourable" cases:

THxxx
xTHxx
xxTHx
xxxTH

where x represent M, A or S.

If i represents the number of "x" at the left of TH, then the number of permutations for each of these 4 cases is:

3! / (3 - i)!

Adding them you get

3! / 0! + 3! / 1! + ·! / 2! + 3! / 3! = 16

So the probability is 16 / 5! = 16 / 120 = 0.13 roughly.

Now, don't take this too seriously, I did it for fun and it may be wrong...

Written on June 5th:
IT IS wrong, see zaza's post below.

[zaza]

You also have TxHxx, TxxHx, xTxHx and so on. You need to consider the four possible positions of T (not the last, obviously, because then the H can't be to the right of it) and then the number of positions that are left for the H to occupy. i.e., there are four positions for H if T is in position 1 etc. Then each of these you multiply by the number of permutations of the other 3 letters.

zaza

[krtxmrtz]

You also have TxHxx, TxxHx, xTxHx and so on. You need to consider the four possible positions of T (not the last, obviously, because then the H can't be to the right of it) and then the number of positions that are left for the H to occupy. i.e., there are four positions for H if T is in position 1 etc. Then each of these you multiply by the number of permutations of the other 3 letters.

zaza

Oh no, I forgot about all these cases! It's unbelievable how these type of problems tend to fool your mind and trick you into what seems correct but isn't. Thanks for pointing it out zaza, maybe I'll work on it again when I have some time.