How many 3-letter words can be formed from the letters of the word 'OBJECTS' if repetition is not allowed?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 840
Note: The given answer is as per the official NTA Answer Key. However, the answer given by NTA appears to be incorrect: 3-letter words from the 7 different letters number \(7\times6\times5=210\), so Option (3) is the correct answer, as explained below. 840 is the count of 4-letter words; NTA’s key added a letter the question never asked for.
A “word” here is any arrangement of letters, so order matters and the count is a permutation. OBJECTS has 7 different letters, and no letter may be used twice.
First letter: any of the 7
Second letter: any of the 6 left
Third letter: any of the 5 left
Number of words: \(^{7}P_3=7\times6\times5=210\)
Option (1) 35 — Incorrect. \(^{7}C_3=35\) counts the selections of 3 letters, without arranging them.
Option (2) 120 — Incorrect. \(6\times5\times4=120\) would be the count for a word of only 6 different letters.
Option (3) 210 — Should have been correct. It is \(^{7}P_3\), the number of 3-letter arrangements of 7 different letters.
Option (4) 840 — Correct as per NTA; however, it counts 4-letter words. \(^{7}P_4=7\times6\times5\times4=840\).
Tip: “Words”, “numbers” and “arrangements” are counted with permutations (order matters); “committees”, “groups” and “selections” with combinations.