Let b1b2b3b4 be a 4-element permutation with bi{1, 2, 3,…..,100} for 1≤i≤4 and bi ≠bj for i≠ j, such that either b1, b2, b3 are consecutive integers or b2, b3, b4 are consecutive integers. Then the number of such permutations b1b2b3b4 is equal to _______ .
Correct Answer: 18915
Solution and Explanation
98 sets of three consecutive integer and 97 sets of four consecutive integers.
By Using the principle of inclusion and exclusion,
The number of permutations of b1b2b3b4 = The number of permutations when b1b2b3 are consecutive + The number of permutations when b2b3b4 are consecutive – The number of permutations when b1b2b3b4 are consecutive.
=97 × 98 + 97 × 98 – 97 = 97 × 195
= 18915
So, the answer is 18915.
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Concepts Used:
Permutations and Combinations
Permutation:
Permutation is the method or the act of arranging members of a set into an order or a sequence.
- In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point.
- A permutation is used in many events of daily life. It is used for a list of data where the data order matters.
Combination:
Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.
- Combination refers to 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.