Loading...
The letters D, G, I, I , and T can be used to form 5-letter strings as DIGIT or DGIIT. Using these letters, how many 5-letter strings can be formed in which the two occurrences of the letter I are separated by at least one other letter?
Let's understand what we're looking for. We have 5 letters: D, G, I, I, and T. Notice that we have two I's and the other letters appear once each.
We want to arrange these 5 letters into strings where the two I's are NOT next to each other - they must be "separated by at least one other letter." This means there must be at least one letter between the two I's.
For example:
Process Skill: TRANSLATE - Converting the constraint "separated by at least one other letter" into "not adjacent"
First, let's find how many ways we can arrange all 5 letters without any restrictions.
If all letters were different, we'd have \(5! = 120\) arrangements. However, since we have two identical I's, we need to account for this repetition.
When we have identical objects, we divide by the number of ways to arrange the identical items among themselves. The two I's can be arranged in \(2! = 2\) ways, but since they're identical, these arrangements look the same.
Total arrangements = \(\frac{5!}{2!} = \frac{120}{2} = 60\)
Now let's count how many arrangements have the two I's next to each other (adjacent).
We can think of the two adjacent I's as a single "block" or unit. So instead of arranging D, G, I, I, T, we're arranging D, G, [II], T - that's 4 units.
These 4 units can be arranged in \(4! = 24\) ways.
Let's verify with a few examples:
So there are 24 arrangements where the I's are adjacent.
To find arrangements where I's are NOT adjacent, we subtract the adjacent arrangements from the total arrangements.
Arrangements where I's are separated = Total arrangements - Adjacent arrangements
Arrangements where I's are separated = \(60 - 24 = 36\)
Process Skill: APPLY CONSTRAINTS - Using complementary counting to handle the "at least one letter between" constraint efficiently
The number of 5-letter strings where the two I's are separated by at least one other letter is 36.
This matches answer choice (D) 36.
Faltering Point 1: Misinterpreting the constraint "separated by at least one other letter"
Students often confuse this phrase and think it means the I's must have exactly one letter between them, rather than understanding it means "not adjacent" (one OR more letters between them). This leads to a completely different counting approach where they try to place exactly one letter between the I's.
Faltering Point 2: Forgetting to account for identical letters in permutation calculations
When calculating total arrangements, students frequently use \(5! = 120\) directly, forgetting that having two identical I's means we must divide by 2! to avoid overcounting. This gives an incorrect total of 120 instead of 60, making all subsequent calculations wrong.
Faltering Point 3: Attempting direct counting instead of using complementary counting
Students often try to directly count arrangements where I's are separated by considering all possible positions, which becomes complex and error-prone. They miss that complementary counting (Total - Adjacent) is much simpler and more reliable for this type of constraint.
Faltering Point 1: Incorrect calculation when treating adjacent I's as a block
When using the "block method" to count adjacent arrangements, students sometimes forget that the two I's within the block are identical, so they don't need to multiply by 2!. The correct count is simply \(4! = 24\) arrangements of the 4 units (D, G, [II], T).
Faltering Point 2: Arithmetic errors in the complementary counting subtraction
Students may correctly calculate total arrangements (60) and adjacent arrangements (24) but make simple arithmetic mistakes when subtracting: 60 - 24. Some might accidentally add instead of subtract, or make computational errors leading to incorrect final results.
Faltering Point 3: Errors while selecting the answer
No likely faltering points - the final calculation directly gives a single numerical result that matches one of the answer choices, with no additional interpretation or selection criteria needed.