MEGA 2560 with 48 relays

"5V 8-Channel Relay interface board, and each one needs 15-20mA Driver Current "

Yes, channels can be driven from Arduino pins. Low will drive the relay.
Channels have current limit resistors built in already.

Mega can sink a max of 800mA at one time, therefor 40 relays max on at one time.
Read the 2560 data sheet, pg 368-369, to understand how to spread out the loads. Section 3 below applies to you:

  1. Although each I/O port can sink more than the test conditions (20mA at VCC = 5V, 10mA at VCC = 3V) under steady state
    conditions (non-transient), the following must be observed:
    ATmega1281/2561:
    1.)The sum of all IOL, for ports A0-A7, G2, C4-C7 should not exceed 100mA.
    2.)The sum of all IOL, for ports C0-C3, G0-G1, D0-D7 should not exceed 100mA.
    3.)The sum of all IOL, for ports G3-G5, B0-B7, E0-E7 should not exceed 100mA.
    4.)The sum of all IOL, for ports F0-F7 should not exceed 100mA.
    ATmega640/1280/2560:
    1.)The sum of all IOL, for ports J0-J7, A0-A7, G2 should not exceed 200mA.
    2.)The sum of all IOL, for ports C0-C7, G0-G1, D0-D7, L0-L7 should not exceed 200mA.
    3.)The sum of all IOL, for ports G3-G4, B0-B7, H0-B7 should not exceed 200mA.
    4.)The sum of all IOL, for ports E0-E7, G5 should not exceed 100mA.
    5.)The sum of all IOL, for ports F0-F7, K0-K7 should not exceed 100mA.
    If IOL exceeds the test condition, VOL may exceed the related specification. Pins are not guaranteed to sink current greater
    than the listed test condition.

  2. Although each I/O port can source more than the test conditions (20mA at VCC = 5V, 10mA at VCC = 3V) under steady
    state conditions (non-transient), the following must be observed:
    ATmega1281/2561:
    1)The sum of all IOH, for ports A0-A7, G2, C4-C7 should not exceed 100mA.
    2)The sum of all IOH, for ports C0-C3, G0-G1, D0-D7 should not exceed 100mA.
    3)The sum of all IOH, for ports G3-G5, B0-B7, E0-E7 should not exceed 100mA.
    4)The sum of all IOH, for ports F0-F7 should not exceed 100mA.
    ATmega640/1280/2560:
    1)The sum of all IOH, for ports J0-J7, G2, A0-A7 should not exceed 200mA.
    2)The sum of all IOH, for ports C0-C7, G0-G1, D0-D7, L0-L7 should not exceed 200mA.
    3)The sum of all IOH, for ports G3-G4, B0-B7, H0-H7 should not exceed 200mA.
    4)The sum of all IOH, for ports E0-E7, G5 should not exceed 100mA.
    5)The sum of all IOH, for ports F0-F7, K0-K7 should not exceed 100mA.
    If IOH exceeds the test condition, VOH may exceed the related specification. Pins are not guaranteed to source current
    greater than the listed test condition.