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Below isn't a GMAT-style question, but I think variations on this type of question (number of balls, number of colours) would be good practice for combinations. I just need to understand this one first before I can learn variations!
Let’s say there are 32 balls in a bucket. Each ball is one of 4 colours (blue, green, red, yellow) so that there are 8 balls of each colour. Within each colour the balls are labelled from numbers 1 through 8. How many combinations of balls are there? Split these into types (e.g. 5567) and subtypes (e.g. 5b5g6r7y).
I get tripped up when I try to sum each 'type' and 'subtype' of combination.
I don't know whether I'm missing subtypes and/or simply just have bad math (I'm leaning towards just the latter). The "NO repeats" one is definitely wrong as the subtypes don't sum to the total.
0. TOTAL combinations: 32C4=35,960
1. FOUR of a number (e.g. 5555): 4C4*8=8
2. THREE of a number (e.g. 5556): 4C3*8*28=896
2a. with a shared colour: 4C3*8*7*3=672
2b. with different colour: 4C3*8*7=224
3. TWO numbers repeating (e.g. 5566): 8C2*4C2*4C2=1008
3a. with different colours: 8C2*4C2=168
3b. one matching colour: 8C2*4C2*2*2=672
3c. two matching colours: 8C2*4C2=168
4. ONE repeat of a number (e.g. 5567): 4C2*8*28*24/2=16,128
4a. with different colours: 4C2*8*7*6=2016
4b. with non repeating numbers of matching colour: 4C2*8*7C2*2=2016
4c. with a repeating number matching colour with a non-repeating number: 4C2*8*7*2*6*2=8064
4d. two matching colours: 4C2*8*14*6/2=2016
4e. three matching colours:4C2*8*7*6=2016
5. NO repeats of a number (e.g. 5678): 32*28*24*20/4!=17,960
5a. with different colours: 8*7*6*5=1680
5b. with one matching colour: 8C2*4*16*10/2=10,080
5c. with two matching colours: 8C2*4*6C2*3=5040
5d. with three matching colours: 8C3*4*15=3360
5e. with four matching colours: 8C4*4=280
Let me know if you want an explanation for any of the above numbers
Let’s say there are 32 balls in a bucket. Each ball is one of 4 colours (blue, green, red, yellow) so that there are 8 balls of each colour. Within each colour the balls are labelled from numbers 1 through 8. How many combinations of balls are there? Split these into types (e.g. 5567) and subtypes (e.g. 5b5g6r7y).
I get tripped up when I try to sum each 'type' and 'subtype' of combination.
I don't know whether I'm missing subtypes and/or simply just have bad math (I'm leaning towards just the latter). The "NO repeats" one is definitely wrong as the subtypes don't sum to the total.
0. TOTAL combinations: 32C4=35,960
1. FOUR of a number (e.g. 5555): 4C4*8=8
2. THREE of a number (e.g. 5556): 4C3*8*28=896
2a. with a shared colour: 4C3*8*7*3=672
2b. with different colour: 4C3*8*7=224
3. TWO numbers repeating (e.g. 5566): 8C2*4C2*4C2=1008
3a. with different colours: 8C2*4C2=168
3b. one matching colour: 8C2*4C2*2*2=672
3c. two matching colours: 8C2*4C2=168
4. ONE repeat of a number (e.g. 5567): 4C2*8*28*24/2=16,128
4a. with different colours: 4C2*8*7*6=2016
4b. with non repeating numbers of matching colour: 4C2*8*7C2*2=2016
4c. with a repeating number matching colour with a non-repeating number: 4C2*8*7*2*6*2=8064
4d. two matching colours: 4C2*8*14*6/2=2016
4e. three matching colours:4C2*8*7*6=2016
5. NO repeats of a number (e.g. 5678): 32*28*24*20/4!=17,960
5a. with different colours: 8*7*6*5=1680
5b. with one matching colour: 8C2*4*16*10/2=10,080
5c. with two matching colours: 8C2*4*6C2*3=5040
5d. with three matching colours: 8C3*4*15=3360
5e. with four matching colours: 8C4*4=280
Let me know if you want an explanation for any of the above numbers
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