NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

The number of bit strings of length 10 that contain either five consecutive 0's or five consecutive 1's is:

Choose the correct answer:

Explanation

1. Calculate ∣A∣ (Strings with five consecutive 0's) A bit string of length $n = 10$ with at least $k = 5$ consecutive 0's can be calculated using a recurrence relation or by manual counting of start positions, but for $n = 2 k$ , a common simplified approach is used. For $n = 10$ and $k = 5$ , the number of such strings is 112. (Due to symmetry, $∣ B ∣ = ∣ A ∣ = 112$ ).

2. Calculate $∣ A \cap B ∣$ (Strings with five consecutive 0's AND five consecutive 1's) For a string of length 10 to have both five consecutive 0's and five consecutive 1's, it must consist of exactly one block of five 0's and one block of five 1's. The possible patterns are:

0000011111 (1 string)
1111100000 (1 string)

So, $∣ A \cap B ∣ = 2$ .

3. Apply Inclusion-Exclusion Principle

|A \cup B| = |A| + |B| - |A \cap B|

|A \cup B| = 112 + 112 - 2

|A \cup B| = 224 - 2

|A \cup B| = 222

Correct Option: (d)