Discrete Maths (Degree level)

Hi, I have a list of 20 questions to do but im not sure about these 4?

2.
How many 5 letter 'words' can be made from the Roman alphabet
(i) if repetitions are allowed
(ii) if repetitions are not allowed

3.
(i) A binary string consists of a sequence of 0s and 1s. How many binary strings are there of length 7?
(ii) A ternary string consists of a sequence of 0s and 1s. How many ternary strings are there of length 7?

4.
(i) How many 'anagrams' are there of the letters in the word MATHS?
(ii) How many 'anagrams' are there of the word MATHEMATICS?

5.
How many three-digit integers (base 10) have no repeated digit?

Thanks so so much

Reply 1

jpowell

manps

2(i)

A word is a sequence of letters. There are 26 letters in our alphabet, we are basically choosing 5 letters from a bag with infinitely many letters. There are 26^5 possible words.

2(ii)

Here we are choosing letters from a bag with only 26 letters, one of each, because repetitions are not allowed. So there are 26*25*24*23*22*21 or 26!/(26-5)!

3(i) 3^7
3(ii) assuming you meant tenary as a sequence of 0's, 1's and 2's then 3^7

4(i) The question is asking how many possible arrangements are there of 5 distinct objects. Answer 5!

The general number of permutations of objects when you have repeated objects i.e. 7 objects in total with 3 the same is 7!/3!, and if you had 7 objects of which 2 lots of 3 were similar then it would be 7!/3!3!

i.e. x total objects with a,b,c... being the number of distinct repeated objects is x!/a!b!c!....

So MATHEMATICS has 11 objects and 3 letters appear twice so the answer is 11!/2!2!2! = 11!/6

5

This is asking how many three digit numbers are there with no number repeated. This is the same as asking how many possible combinations of 3 objects chosen from 10 distinct objects, with the additional condition that the first object cannot be 0 (because then the number would be two digit).

So there are 9 choices for the first digit (1-9) then 9 choices for the second digit (0-9 minus the first digit) and then 8 choices for the third.

So the total number is 9*9*8.