Q75 / 180 · COMEDK 2023 (Morning Shift) · Mathematics
A five-digits number is formed by using the digits $1,2,3,4,5$ with no repetition. The probability that the numbers 1 and 5 are always together, is
- A$\dfrac{1}{5}$
- B$\dfrac{1}{4}$
- Correct answerC$\dfrac{2}{5}$
- D$\dfrac{3}{5}$
Solution
The total number of ways to form a five-digit number using the digits $1, 2, 3, 4, 5$ without repetition is $5! = 120$.
To find the number of ways where $1$ and $5$ are always together, treat the pair $(1, 5)$ as a single unit. This leaves us with $4$ units: $\{1, 5\}, 2, 3, 4$.
The number of ways to arrange these $4$ units is $4! = 24$.
Within the unit $\{1, 5\}$, the digits $1$ and $5$ can be arranged in $2! = 2$ ways (either $15$ or $51$).
Therefore, the total number of favorable arrangements is $4! \times 2! = 24 \times 2 = 48$.
The probability is the ratio of favorable arrangements to total arrangements:
$P = \dfrac{48}{120} = \dfrac{2}{5}$.
Answer: $\dfrac{2}{5}$
To find the number of ways where $1$ and $5$ are always together, treat the pair $(1, 5)$ as a single unit. This leaves us with $4$ units: $\{1, 5\}, 2, 3, 4$.
The number of ways to arrange these $4$ units is $4! = 24$.
Within the unit $\{1, 5\}$, the digits $1$ and $5$ can be arranged in $2! = 2$ ways (either $15$ or $51$).
Therefore, the total number of favorable arrangements is $4! \times 2! = 24 \times 2 = 48$.
The probability is the ratio of favorable arrangements to total arrangements:
$P = \dfrac{48}{120} = \dfrac{2}{5}$.
Answer: $\dfrac{2}{5}$
Hence, the correct answer is option: 'C'.