The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is _________.
Correct Answer: 56
Solution and Explanation
First we arrange 5 red cubes in a row and assume x1, x2, x3, x4, x5 and x6 number of blue cubes between them
Here we see,
x1 + x2 + x3 + x4 + x5 + x6 = 11
also,
x2, x3, x4, x5 ≥ 2
Hence,
x1 + x2 + x3 + x4 + x5 + x6 = 3
Therefore ,
Number of solutions found = 8C5
= 56
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Concepts Used:
Combinations
The method of forming subsets by selecting data from a larger set in a way that the selection order does not matter is called the combination.
- It means the combination of about ‘n’ things taken ‘k’ at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.
- For example, Imagine you go to a restaurant and order some soup.
- Five toppings can complement the soup, namely:
- croutons,
- orange zest,
- grated cheese,
- chopped herbs,
- fried noodles.
But you are only allowed to pick three.
- There can be several ways in which you can enhance your soup with savory.
- The selection of three toppings (subset) from the five toppings (larger set) is called a combination.
Use of Combinations:
It is used for a group of data (where the order of data doesn’t matter).