Hello! I am studying Arduino and I have a multi-LED control exercise possible at the same time with Arduino UNO R3 board.
Has anyone done this exercise yet? If anyone done it so can you help me solving this solution.
Thanks a lot for reading my question and help me solving this solution. 
???
Do you know how to control one LED? What does that need?
Once you know that just look how many of whatever is needed your board has and there's your answer.
Steve
Perhaps you should first explain - in detail - what "a multi-LED control exercise" actually means to you, as it means little to us? 
How many LEDs can be controlled with only one circuit board Arduino UNO R3
Can and should, are two different questions.
If you want several LEDs to be controlled from the same pin, the answer is practically as many as you want providing you keep the current down to what the Arduino can handle. That is commonly thought to be 20mA per pin. However, you generally want enough current for each LED to be visible and so that depends on the exact specification of the LED. Some LEDs can be visible when given a current of 0.1 mA or less, where as others might need 54 to 10mA to see them.
What ever the current capabilities of a single pin, the pins are grouped into ports and each port or combination of ports has its own limits, in the data sheet it says these are:-
Although each I/O port can source more than the test conditions (20 mA at VCC = 5V, 10 mA at VCC = 3V) under steady state conditions (non-transient), the following must be observed:
ATmega48PA/88PA/168PA/328P:
1] The sum of all IOH, for ports C0 - C5, D0- D4, ADC7, RESET should not exceed 150 mA.
2] The sum of all IOH, for ports B0 - B5, D5 - D7, ADC6, XTAL1, XTAL2 should not exceed 150 mA.
If IIOH exceeds the test condition, VOH may exceed the related specification. Pins are not guaranteed to source current greater than the listed test condition.
what is more the DC Current VCC and GND Pins must not exceed a total of 200.0 mA or the Arduino will be damaged. Remember this means the current for all the LEDs plus the current to make the chip operate, this is about 20mA for the 16MHz clock of the Arduino. ( Note no actual figures are in the data sheet for this frequency of operation ).
So without knowing more about these LEDs your question is impossible to answer.
If the question is just how many LEDs can be controlled separately directly from the Arduino then the trivial answer is however many pins the Arduino has.