I will form a random 5 digit number using only the integers 1 through 7 and allowing no integer to be used mor

I will form a random 5 digit number using only the integers 1 through 7 and allowing no integer to be used more than once in any 5 digit number. Find the probability that one of the 5 digit number has exactly one two of the integers 1, 2, or 3 (and the rest bigger than 3)I will form a random 5 digit number using only the integers 1 through 7 and allowing no integer to be used mor
Ok, your question is worded a little oddly, but I think you mean to say the probability that it has exactly one or two of the integers 1,2,3. Meaning, it can't have all 3.





Well, first find the probability that it has all 3. Find the number of 5 digit numbers using only 1-7 that include 1,2,3.


Choose a spot for 1 in 5 ways, choose a spot for 2 in 4 ways, choose a spot for 3 in 3 ways. You have two positions left to fill with 4,5,6, or 7. Choose a number for the first position in 4 ways, and finally a number for the last position in 3 ways. So there are 5*4*3*4*3 total.





Now there are 7*6*5*4*3 total possible arrangements, so the probability that it includes 1,2 and 3 is: (5*4*3*4*3)/(7*6*5*4*3) = 4*3/7*6 = 2/7.





So the probability that it does not contain all of 1,2,3 is: 1 - 2/7 or 5/7.