Question

The letters of the word LOGARITHM are arranged at random, the probability that exactly 4 letters are between G and H is

• A. $\frac{1}{12}$
• B. $\frac{5}{12}$
• C. $\frac{1}{9}$
• D. $\frac{1}{4}$

Answer 1

Answer: (C)

$\frac{1}{9}$

Answer 2

The problem asks for the probability that exactly 4 letters are between G and H when the letters of the word LOGARITHM are arranged at random.

1. Total number of arrangements:

The word LOGARITHM has 9 distinct letters. The total number of ways to arrange these 9 distinct letters is 9!.

Total arrangements = $9! = 362,880$.

2. Number of favorable arrangements:

We want exactly 4 letters to be between G and H.

• Step 1: Identify the number of possible positions for the block (G _ _ _ _ H).

  Let N be the total number of letters (9). Let k be the number of letters between G and H (4). The block (G _ _ _ _ H) has $k+2 = 4+2=6$ letters. The number of possible starting positions for G (or H) such that there are 4 letters between them is $N - (k+1) = 9 - (4+1) = 9 - 5 = 4$. These are (G at 1, H at 6), (G at 2, H at 7), (G at 3, H at 8), (G at 4, H at 9). So, 4 pairs of positions.

• Step 2: Arrange G and H.

  For each pair of positions, G and H can be arranged in $2!$ ways.

• Step 3: Arrange the remaining letters.

  The remaining $(N-2)$ letters (7 letters) can be arranged in the remaining $(N-2)$ positions (7 positions) in $7!$ ways.

Favorable arrangements = (Number of position pairs) $\times$ (Arrangement of G and H) $\times$ (Arrangement of remaining letters)

Favorable arrangements = $4 \times 2! \times 7!$

Favorable arrangements = $4 \times 2 \times 5040 = 8 \times 5040 = 40,320$.

3. Calculate the probability:

Probability = (Favorable arrangements) / (Total arrangements)

Probability = $\frac{4 \times 2! \times 7!}{9!}$

Probability = $\frac{4 \times 2 \times 7!}{9 \times 8 \times 7!}$

Probability = $\frac{4 \times 2}{9 \times 8}$

Probability = $\frac{8}{72}$

Probability = $\frac{1}{9}$