What is the difference between a multiple pulley system in which the string you pull on comes down from the top pulley and the one in which the string you pull on comes up from the bottom pulley?
When the string you pull on comes down from the top pulley, it doesn’t exert its tension on the thing being lifted so it doesn’t count when add up the strings. But when the string you pull on comes up from the bottom pulley, that string is also helping to lift the object. That string does count. Thus if the multiple pulley has 5 segments going up and down between the two pulleys and one more segment going up to your hand, the total number of segments lifting the object is 6 and that object experiences an upward force equal to 6 times the tension in the string.