You are thinking of using some leftover ceramic tiles to make a decorative border for a patio. You have ten blue, four gold, and six white tiles. In how many ways can you make the border if it requires 15 tiles and you use only blue and white ones? Explain your reasoning.
i need help on this question
number of ways to choose:
6W,9B
5W,10B
Number of ways to arrange 6W,9B = (15!/6!9!) = 5005
number of ways to arrange 5W,10B = 15!/(5!10!) = 3003
total number of ways = 5005+3003 = 8008
To calculate the number of ways to make the border using only blue and white tiles, we can use the concept of combinations.
First, let's determine the number of blue tiles we can use for the border. Since we have ten blue tiles, we can use anywhere from 0 to 10 blue tiles for the border.
Now, let's consider the number of white tiles we can use. Since the border requires a total of 15 tiles and we have already considered the possibility of using all the blue tiles, we need to use the remaining tiles. Therefore, the number of white tiles to be used would be 15 minus the number of blue tiles used.
For each combination of blue tiles used, we can calculate the corresponding number of white tiles used and hence determine how many ways we can make the border.
Let's go through each possibility step-by-step:
1. If we use 0 blue tiles, we need all 15 tiles to be white. There is only one way to do this.
2. If we use 1 blue tile, we would need 14 white tiles. As we have 6 white tiles available, the number of ways to arrange these would be the combination of choosing 14 out of 6. This can be calculated as 6 choose 14, denoted by (6 C 14) or C(6, 14).
Similarly, we can calculate the number of ways for the following cases:
3. Using 2 blue tiles: (7 C 13)
4. Using 3 blue tiles: (8 C 12)
5. Using 4 blue tiles: (9 C 11)
6. Using 5 blue tiles: (10 C 10)
Finally, we sum up all these possibilities to get the total number of ways to make the border:
Total = (0 C 15) + (6 C 14) + (7 C 13) + (8 C 12) + (9 C 11) + (10 C 10)
Now, we can calculate each of these combinations:
(0 C 15) = 1
(6 C 14) = 6! / (14! * (6-14)!) = 6
(7 C 13) = 7! / (13! * (7-13)!) = 7
(8 C 12) = 8! / (12! * (8-12)!) = 70
(9 C 11) = 9! / (11! * (9-11)!) = 36
(10 C 10) = 1
Total = 1 + 6 + 7 + 70 + 36 + 1 = 121
Therefore, there are 121 ways to make the border using only blue and white tiles.
To solve this problem, you need to determine the number of ways you can arrange the blue and white tiles to create a border with a total of 15 tiles. Since you have 10 blue and 6 white tiles, you can use the concept of combinations to determine the number of ways to choose the number of blue tiles for the border.
First, let's consider if we were to use all 10 blue tiles in the border. In this case, we would have no white tiles left. However, we need a total of 15 tiles, so this arrangement is not possible.
Next, let's consider using 9 blue tiles. Since we have 10 blue tiles in total, we can choose 9 in a total of C(10, 9) ways. This means we have C(10, 9) = 10 ways to choose 9 blue tiles.
Similarly, we can proceed with using 8, 7, and so on blue tiles, until we have a total of 15 tiles.
Now, let's calculate the number of ways for each case:
- If we use 8 blue tiles, we have C(10, 8) = 45 ways.
- If we use 7 blue tiles, we have C(10, 7) = 120 ways.
- If we use 6 blue tiles, we have C(10, 6) = 210 ways.
- If we use 5 blue tiles, we have C(10, 5) = 252 ways.
- If we use 4 blue tiles, we have C(10, 4) = 210 ways.
- If we use 3 blue tiles, we have C(10, 3) = 120 ways.
- If we use 2 blue tiles, we have C(10, 2) = 45 ways.
- If we use 1 blue tile, we have C(10, 1) = 10 ways.
Finally, we sum up the number of ways for each case since they are all independent events:
10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 = 1022.
Therefore, there are 1022 ways to make the border using only blue and white tiles.