What is the hydronium ion concentration in a solution prepared by mixing 50.00 mL of 0.10 M HCN with 50.00 mL of 0.020 M NaCN?... ?

HCN ---> Weak acid

NaCN ---> Salt of a weak acid (specifically, a salt of HCN). This is the base that is referred to below.

This type of solution is called a buffer and these type of problems are solved using the Henderson-Hasselbalch Equation:

pH = pKa + log [base/acid]

pKa = -log 4.9 × 10^-10 = some value you should determine.

You need to know either the molarities of the base and the acid or the total moles of each. I'll use molarities.

the acid is diluted from a total volume of 50 mL to 100 mL. Its molarity goes from 0.10 to 0.05.

the base is also diluted from 50 to 100 mL. Its molarity goes from 0.01 to 0.01

So, now we can solve:

pH = ___ + log (0.01 / 0.05)

At the ___ will go the pKa value you calculated.

After you have the pH, then use 10^-pH to get the [H^+]

Notice the 0.01 / 0.05 ratio and then, contemplate this ratio:

0.02 / 0.1

Yes, they are the same. If I ignored the dilutions, the pH would be the same. This is a special case because the initial volumes are equal. If 50 mL of one was mixed with 25 mL of the other (or any other no-equal combo), then you'd have to calculate the new molarities.

I will leave you to examine how the calculation might be done with moles.

I have lots of solved buffer problems here:

https://www.chemteam.info/AcidBase/Buffer-Examples...

pH of your solution can be calculated using the Henderson equation below 

pH = -log( 4.9 × 10-10 ) + log ( 0.020 / 0.10) = 9.31 -0.70

pH=  8.61

[H+] = 10 ^-8.61 = 2.4 *10 ^-9

my solution agrees with answer a