Arithmetic Grade Four

Lesson One hundred and sixty: Combinations

Let's say that Jill wakes up one morning and must decide what to wear that day. Let's say that she has two pairs of shoes ( one white and one black ), 2 pairs of pants ( one black and one blue ), and three shirts ( one red, one green, and one yellow ) that are clean and ready for her to wear. How many different combinations of these items does she have to choose from? 


      SHOES	  PANTS		SHIRTS		COMBINATIONS
                      /---------red		white, black, red
            -----black ---------green		white, black, green
           /          \---------yellow          white, black, yellow
      white
           \          /---------red		white, blue, red
            -----blue ----------green		white, blue, green
                      \---------yellow		white, blue, yellow

                      /---------red		black, black, red
            -----black ---------green		black, black, green
           /          \---------yellow          black, black, yellow
      black
           \          /---------red		black, blue, red
            -----blue ----------green		black, blue, green
                      \---------yellow		black, blue, yellow
You can count up the combinations and see that there are a total of 12 possible combinations. Drawing out a tree diagram like this can be a time consuming and so it is good to know that there is a short cut for finding possible combinations: 

             SHOES	PANTS	SHIRTS	COMBINATIONS
               2    x     2   x    3   =   12
Let's say that the next day Jill had more items to choose from: 3 pairs of shoes, 4 pairs of pants, and 4 shirts. How many combinations would she have now? 

             SHOES	PANTS	SHIRTS	COMBINATIONS
               3    x     4   x    4   =   48

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