What's the difference of placing multiple resistors and one resistor for limiting the current of LEDs?
What you're suggesting will work as long as only one LED is lit at a time. But as soon as you light more than one LED at once, the current from two LEDs will flow through one resistor. This will cause twice the voltage drop, and both LEDs will be dim. Dimmer still if all three are on at once.

You will need to wire it as shown, with three resistors.