Help ASAP! I have a test tomorrow! - combinatorics (1 Viewer)

[tk8]

Hi! I've read a bunch of worked solutions for the attached question, but I just can't understand why you do 10x4x9=360 for the case of a single digit and triple digit.

Any help will be appreciated

(btw the answer is A)

 

[cossine]

Altogether there are 10000 codes.

let x represent digit 1 that is repeating. Let y represent digit 2 that is repeating.

So the number of permutations of xxyy is:

4! / (2!*2!) = 6 # This is 4C2 or 4 choose 2.

There are 10 possible digits x could take and 9 possible digits y could take.

So altogether 540 ways for two repeating digits.

# Divide by 2 because when x = 0, y = 1 => 0011(xxyy) is the same as when y =1 , x =0 => 0011 (yyxx)

So actual number of permutations is 270.



Similarly the number of permutations of xxxy is:

4.

There are 10 possible digits x could take and 9 possible digits y could take.

So the number of total outcomes is 4*10*9 = 360



360 + 270 = 630


So to answer your question it come froms the rule of product axioms.

https://en.wikipedia.org/wiki/Rule_of_product

 

[tk8]

Altogether there are 10000 codes.

let x represent digit 1 that is repeating. Let y represent digit 2 that is repeating.

So the number of permutations of xxyy is:

4! / (2!*2!) = 6 # This is 4C2 or 4 choose 2.

There are 10 possible digits x could take and 9 possible digits y could take.

So altogether 540 ways for two repeating digits.

# Divide by 2 because when x = 0, y = 1 => 0011(xxyy) is the same as when y =1 , x =0 => 0011 (yyxx)

So actual number of permutations is 270.



Similarly the number of permutations of xxxy is:

4.

There are 10 possible digits x could take and 9 possible digits y could take.

So the number of total outcomes is 4*10*9 = 360



360 + 270 = 630


So to answer your question it come froms the rule of product axioms.

https://en.wikipedia.org/wiki/Rule_of_product

Thanks for replying! Could I just ask, for the first case of xxyy, why do we only do 10x9? Why don't we do 10x10x9x9 because there are 2 x's and 2 y's?

 

[cossine]

Thanks for replying! Could I just ask, for the first case of xxyy, why do we only do 10x9? Why don't we do 10x10x9x9 because there are 2 x's and 2 y's?

So x, y represent some unknown digit.

There 10 possible values x could take.

So if x =9, and y = 5

we would have (for the permutation xxyy)

9955.

So the second character depends on the first character that is 9. Similarly the fourth character depends on the third character that is 5.

That why we have 9*10