a tree is 67 inches tall. how tall is it in feet and inches?
- If in a bit’s string of {0,1}, of length 4, such that no two ones are together. Then the total number of such possible strings are?
a) 1
b) 5
c) 7
d) 4
The Correct Answer and Explanation is:
Part 1: Converting 67 Inches into Feet and Inches
To convert 67 inches into feet and inches:
- There are 12 inches in a foot.
- Divide 67 by 12:
67÷12=5 remainder 767 \div 12 = 5\text{ remainder }7
So, 67 inches = 5 feet and 7 inches.
Part 2: Binary Strings of Length 4 with No Two Adjacent 1s
We are asked to find how many bit strings of length 4 made up of {0,1} do not have two 1s next to each other.
Step-by-step Explanation:
We need to count all strings of length 4 such that no two 1s are adjacent.
Let’s define:
- ana_n: number of valid strings of length nn with no two consecutive 1s.
This is a classic recurrence relation problem. The recurrence relation is: an=an−1+an−2a_n = a_{n-1} + a_{n-2}
Where:
- a1=2a_1 = 2: “0”, “1”
- a2=3a_2 = 3: “00”, “01”, “10”
Using the recurrence:
- a3=a2+a1=3+2=5a_3 = a_2 + a_1 = 3 + 2 = 5
- a4=a3+a2=5+3=8a_4 = a_3 + a_2 = 5 + 3 = 8
So there are 8 such strings of length 4 in total that do not have consecutive 1s.
But wait! The question is more specific — it wants strings that contain 1s, but no two 1s are together. So we need to exclude the string with no 1s at all (i.e., “0000”).
From the 8 valid strings with no adjacent 1s:
- Valid strings:
0000,0001,0010,0100,0101,1000,1001,1010
Only 7 of these contain at least one ‘1’ and no adjacent 1s.
✅ Final Answer: (c) 7
There are 7 valid bit strings of length 4 that contain at least one ‘1’ and no two 1s are adjacent.
