how many different ways can the letters PROBLEM be arranged? Considering all of the possibles arrangements of the letters, what is the probability of actually s

Question

Need help! 

Answer 1

Hi Lane,

 

How many different ways are there to represent ‘A’   --  1

How many for ‘AB’    --  2   (AB and BA)

How many for 'ABC'   --  (ABC, ACB , BAC, BCA,  CAB, CBA) = 6

Hmm, we put each letter first, then for the rest of the letters we put each letter first, then for the remaining letters, we put each letter first, then …      this is called recursion and it is extremely important in math, computer science, modeling natural phenomena, etc.

So, we have 1,  1*2,  1*2*3,  … combinations.   If there are 7 letters as in our problem, that is 1*2*3*4*5*6*7  (note,  this assumes that the order of duplicate characters matters, we must reduce this number if BB is the same as BB).

What is 1*2*3*4*5*6*7  ?       5040  (PLZ check my math)

Now, only one of those spellings is “PROBLEM”   (there might be more if there were duplicate letters, remember)

    So, the likelihood of selecting “PROBLEM” from this bag of 7-letter words is  1/5040.

 

p.s., at some point, you will learn a general formula for this solution;  be sure to understand (not just memorize) it.