Light leds on this board one at a time
This is your circuit.

I think you can take it from here by looking at the diagram, but please note that e.g. 1 0 0 0 0 will light up 3 and 1, so be careful.

To show more than one, turn on multiple LEDs in rapid succession. It will give the effect that more than one are on. This is known as multiplexing.

Update:

After Googling around, I understood it better. I also wrote this code.

int IO1 = 50;
int IO2 = 48;
int IO3 = 46;
int IO4 = 44;
int IO5 = 41;
int del = 1000;

void setup() {

  pinMode(IO1, INPUT);
  pinMode(IO2, INPUT);
  pinMode(IO3, INPUT);
  pinMode(IO4, INPUT);
  pinMode(IO5, INPUT);
  Serial.begin(9600);
}

// the loop routine runs over and over again forever:
void loop() {
    displayLED(1);
    delay(del);
    displayLED(2);
    delay(del);
    displayLED(3);
    delay(del);
    displayLED(4);
    delay(del);
    displayLED(5);
    delay(del);
    displayLED(6);
    delay(del);
    displayLED(7);
    delay(del);
    displayLED(8);
    delay(del);
    displayLED(9);
    delay(del);
    displayLED(10);
    delay(del);
    displayLED(11);
    delay(del);
    displayLED(12);
    delay(del);
    displayLED(13);
    delay(del);
    displayLED(14);
    delay(del);
    displayLED(15);
    delay(del);
    displayLED(16);
    delay(del);// light up LEDS 1, 2 and 3.
}

void displayLED(int a){
    clear();
    if (a == 1){o(1); o(2); h(1); l(2); Serial.println('1');}
    if (a == 2){o(1); o(2); l(1); h(2); Serial.println('2');}
    if (a == 3){o(1); o(3); h(1); l(3); Serial.println('3');}
    if (a == 4){o(1); o(3); l(1); h(3); Serial.println('4');}
    if (a == 5){o(2); o(3); h(2); l(3); Serial.println('5');}
    if (a == 6){o(2); o(3); l(2); h(3); Serial.println('6');}
    if (a == 7){o(2); o(4); h(2); l(4); Serial.println('7');}
    if (a == 8){o(2); o(4); l(2); h(4); Serial.println('8');}
    if (a == 9){o(1); o(4); h(1); l(4); Serial.println('9');}
    if (a == 10){o(1); o(4); l(1); h(4); Serial.println(10);}
    if (a == 11){o(3); o(4); l(3); h(4); Serial.println(11);}
    if (a == 12){o(3); o(4); h(3); l(4); Serial.println(a);}
    if (a == 13){o(3); o(5); l(3); h(5); Serial.println(a);}
    if (a == 14){o(3); o(5); h(3); l(5); Serial.println(a);}
    if (a == 15){o(4); o(5); l(4); h(5); Serial.println(a);}
    if (a == 16){o(4); o(5); h(4); l(5); Serial.println(a);}

}

void clear(){pinMode(IO1, INPUT);
  pinMode(IO2, INPUT);
  pinMode(IO3, INPUT);
  pinMode(IO4, INPUT);
  pinMode(IO5, INPUT);}

void o(int o){
    if (o == 1){pinMode(IO1, OUTPUT);}
    if (o == 2){pinMode(IO2, OUTPUT);}
    if (o == 3){pinMode(IO3, OUTPUT);}
    if (o == 4){pinMode(IO4, OUTPUT);}
    if (o == 5){pinMode(IO5, OUTPUT);}
}

void h(int l){
    if (l == 1){digitalWrite(IO1, HIGH);}
    if (l == 2){digitalWrite(IO2, HIGH);}
    if (l == 3){digitalWrite(IO3, HIGH);}
    if (l == 4){digitalWrite(IO4, HIGH);}
    if (l == 5){digitalWrite(IO5, HIGH);}
}

void l(int h){
    if (h == 1){digitalWrite(IO1, LOW);}
    if (h == 2){digitalWrite(IO2, LOW);}
    if (h == 3){digitalWrite(IO3, LOW);}
    if (h == 4){digitalWrite(IO4, LOW);}
    if (h == 5){digitalWrite(IO5, LOW);}
}