How would I solve this
#permutations
boreal fossil
terse compassBOT
How would I solve this
- Wait patiently for a helper to come along.
- Once someone helps you, say thank you and close the thread with:
+close
- Feel free to nominate the person for helper of the week in #helper-nominations
- Do not ping the mods, unless someone is breaking the rules.
- If you're happy with the help you got here, and the server overall, you can contribute financially as well:
rugged ivy
Say you have just 1,2,3. How many numbers can you create
boreal fossil
3!
fallen umbra
How would I solve this
I think for (b) there can be more than 1 value of N
boreal fossil
I’m not sure I understand how to do it still this was on my test and I need to correct it I think part A is 5! But don’t understand part b
cunning gale
I’m not sure I understand how to do it still this was on my test and I need to c...
How many digits can a number possibly have (in base 10) if none of the digits are repeated?
cunning gale
it would be 9!?
...no. Think about it. What is the most digits a number in base 10 can have before a digit must be repeated to add a new one?
boreal fossil
oh 10?
cunning gale
Yes.
boreal fossil
9876543210
cunning gale
Right. Now, let's take our condition into account. What is the smallest number of digits a base 10 number can have without repetition where the digits sum to 39?
boreal fossil
987654? so 6 digits
cunning gale
Right. And since every permutation of those digits is a new number, how many numbers is that?
cunning gale
..."multiplied them all"?
boreal fossil
because theres 9 possible digits for the first place then 8 in the next since theres no repeats?
boreal fossil
so its 6!
cunning gale
Yes.
So there are 6! 6-digit numbers that satisfy our condition.
Now let's add a digit, but let's do it smart.
Let's add the digit 0.
Now, bearing in mind 0 cannot be the lead digit, how many numbers does this give us?
boreal fossil
well thers 6 digits the number can start with so would it be 6 x 6! since thew others can be in any order?
cunning gale
Now, adding and subtracting the same amount doesn't change the sum, right?
cunning gale
so 4 can be replaced by 1 and 3?
I was thinking of replacing 4 with 3 and 0 with 1.
boreal fossil
yes so it would then be 7 x 7!
cunning gale
Oh, you are writing down how many numbers we get for each case, right? Because we're going to need to add them all together.
cunning gale
yes so it would then be 7 x 7!
Show your work on that?
boreal fossil
yes
because there are 7 options for the first number 9,8,7,6,5,3,1 and then the other 7 numbers including 0 can be aranged in any order
boreal fossil
but 0 does not add to the total so why can we use 0
cunning gale
but 0 does not add to the total so why can we use 0
Because that would be an 8 digit number. We're still counting 7 digit numbers.
boreal fossil
ok i understand so it would just be 7!
cunning gale
Now, give me a chance to get to a piece of paper, because I think you might have accidentally been right.
boreal fossil
ok lol
cunning gale
Okay, I was wrong. So anyway, 7!. Now let's add 1, subtract 1 again.
cunning gale
What would be the digits?
boreal fossil
9876410
cunning gale
You were right that we subtract 1 from 5. But then we add 1 to 1.
boreal fossil
so it would be 9876432
cunning gale
Right.
boreal fossil
would that be all the 7 digit numbers we can do?
since if 6 was taken out and replaced with 5 then an extra digit would be added making the total to 8
98754321
cunning gale
Yes, actually, but not for that reason.
Because if we subtract 1 from 6 we get 5, and there's nowhere we can add the 1 back without getting a duplicate digit.
Remember. We're subtracting 1 from one digit and adding it to another digit. We aren't "adding it" as a digit.
boreal fossil
So should I continue finding all the combinations and then do I add the final numbers or multiply them?
cunning gale
...we're counting, right?
So if I asked you to count all the balls in a jar, and you counted all the red balls and all the white balls, would you add or multiply the two numbers to get all the balls?
boreal fossil
Add.
boreal fossil
We were moving onto the 8 digits
cunning gale
Right. So we can chuck a 0 onto the end of the two 7-digit sets we have.
What does that give us?
cunning gale
I meant what sets of digits does that give us? You're right, though.
boreal fossil
98764320 and 98765310
cunning gale
Right.
So the 531 one we can't do much with except turn it into the 432 one. What can we do with the 432 one?
boreal fossil
Add a 1 and take away from the 6 to get 5
cunning gale
Which yields?
boreal fossil
8x8!
cunning gale
...no. The set of digits.
boreal fossil
987543210
cunning gale
Where did the 0 come from?
boreal fossil
It was from the 8 digit
cunning gale
Already been over this.
What we subtract from a digit, we add to a different digit, not as a new digit.
cunning gale
How so?
cunning gale
You tell me.
Are there any repeated digits? Is the digit sum what we need it to be?
boreal fossil
Yes it would work
cunning gale
How many numbers does this give us?
boreal fossil
Now there are 3 8 digit numbers
cunning gale
...no. How many permutations of these eight digits?
boreal fossil
8!
cunning gale
Right.
boreal fossil
There is only 1 way for the 9 digits right?
cunning gale
Yeah. Any subtraction addition we do here results in duplication.
boreal fossil
So if I add all the results together that should be the answer
cunning gale
Yeah.
cunning gale
I got 447840
You might've made a mistake. Show your work?
boreal fossil
Yes one second I’ll take a picture
I did those calculations
Then added the results from each
i got a new answer i got 448560
with the working above
cunning gale
i got a new answer i got 448560
This is correct.
boreal fossil
yay thank you for the help
cunning gale
yay thank you for the help
No problem. I know combinatorics can be intimidating, but if you just boil it down to casework and keep your head straight I find that it's relatively intuitive.
I mean, it's basically just counting.