Combination and permutations prove questions pls help

[austere trout]

i dont understand what to do after…

i’m confused

i recently just got introduced this topic

[quaint pagoda]

For 1st image line 3 to 4 you need to multiply first term by k/k (I think you did k!/k!) and multiply 2nd term by n-k (I think you did n-1)

Actually maybe I am misinterpreting what you did but yeah you just need to multiply by k/k and (n-k)/(n-k) to get the denominator to match

For 2nd image Im not sure how you got the first equality since it looks like you just rewrote down the equality after applying the definitions.

For this one you want to force the denominator to go from (n-r)! to (n-(n-r))!. If finding the trick to this is hard, alternatively to prove inequalities you can show that the left side - right side = 0 and this way may be easier to grind out the steps

[austere trout]

i’ve solved the 2nd image

[austere trout]

still confused on the first one tho

[quaint pagoda]

This is good (completely correct idea) but if you want to be slightly more rigorous as a proof you should add <=> at the beginning of each line. You can also start from the last line and go in reverse, adding just => at the start of each line.

This is because you should not start a proof by assuming the statement is true. For example I could prove 1=0 by doing

1 = 0
multiply both sides by 0
0 = 0
Thus 1 = 0

[quaint pagoda]

Did you do the multiply by k/k and (n-k)/(n-k)?

[austere trout]

?

[quaint pagoda]

You seem to be moving the factorial to the end. You should keep it infront of the (n-1)

$(n-1)!k \neq (n-1)k!$

[rocky orchid]

purururuuriuruin

[quaint pagoda]

I think that was actually your original problem but the idea was right

[austere trout]

thank you i’ll fix that up rn

so sorry i’m just new to this topic

[devout tree]

Done?

[austere trout]

yup

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