Problem 85 - Algebra for JEE ADVANCED

Question

A person buys eight packets of TIDE detergent. Each packet contains one coupon, which bears one of the letters of the word TIDE. If he shows all the letters of the word TIDE, he gets one free packet. If he gets exactly one free packet, then the number of different possible combinations of the coupons is a. \({ }^{7} C_{3}\) b. \(\quad{ }^{8} C_{4}\) c. \({ }^{8} C_{3}\) d. \(4^{4}\)

Step-by-Step Solution

Answer

The answer is (a) \[ _{7}C_3 \].

Step 1: Understand the Problem

The problem is about collecting coupons that spell the word "TIDE" to get a free packet. You have 8 packets; each contains one letter of "T", "I", "D", or "E" randomly. To get a free packet of TIDE, you need all four letters. The question asks for the number of ways you can arrange these letters such that you have exactly one full set of the word "TIDE".

Step 2: Calculate Total Ways to Choose the Letters

We need a set of letters where each letter appears at least once (since exactly one complete set is needed to spell TIDE), and we need to figure out the rest of the slots. Consider the remaining four packets after having one "T", one "I", one "D", and one "E." They must complete the other packets without forming another complete set of "TIDE."

Step 3: Determine Remaining Frequency Distribution

After one 'T', 'I', 'D', and 'E' each are used, 4 more letters are to be distributed into these 4 categories such that none of these distributions makes another "TIDE". Essentially, it means choosing 3 out of 4 packets to increase frequency without completing another full sequence.

Step 4: Use Combinatorial Logic

For the remaining distribution of four letters across 'T', 'I', 'D', and 'E', you pick 3 letters or positions among 4 available slots to increase frequency to avoid another complete "TIDE." That is done using the combination formula \[ _{7}C_3 \] which represents choosing 3 additional slots out of 7 possible positions.

Step 5: Verify the Calculation

The ways to distribute exactly one full set include combinations withdrawn from 8 locations maintaining the sequence constraint, further simplifying and verifying to \[ _{7}C_3 \]. Thus option (a) is right.

Key Concepts

Probability

Probability helps us understand the chance or likelihood of a particular event occurring. In the context of the problem, the event is collecting exactly one full set of coupons spelling "TIDE" from eight detergent packets. Each packet contains a single random letter from "T", "I", "D", or "E".

To determine the probability of specific outcomes, we count the ways certain events can occur and compare them to the total number of possible outcomes. Here is how probability applies to this exercise:

• We need each letter in "TIDE" (i.e., "T", "I", "D", "E") at least once to get a free packet, forming a complete set.
• The challenge is to understand how many ways we can distribute the remaining letters in such a way that does not form another "TIDE" set in the remaining packets.
• The probability approach would consider how frequently this setup can be achieved compared to all possible distributions of letters across the packets.

By keeping track of how these letters are distributed among the packets, and understanding the different ways the outcomes can arise, we essentially break down the problem into simpler, more manageable parts.

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics that help us figure out the number of ways to arrange or choose items. In this problem, we are interested in combinations because we care about selecting letters without considering their order.

Here's how the concept applies here:

• Combinations: Used when the order does not matter, as in choosing letters for our coupon collection.


• Once we have one complete set of "TIDE", we must choose how to distribute the rest of the letters across the packets.


• We employ the combination formula \[{_nC_r}\], where \[n\] is the total number of items to choose from, and \[r\] is the number of items to choose. For this exercise, \[{_7C_3}\], which is choosing 3 additional positions to fill among 7 remaining slots, after already having one full set in place.

By using combinations, we efficiently calculate the number of different feasible scenarios that meet the exercise condition of having exactly one set of the word "TIDE".

Coupon Collector Problem

The coupon collector's problem is a classic problem in probability theory. It involves collecting a complete set of different items from some repeated random process. Here, collecting coupons with letters is akin to collecting different types of coupons needed to win a prize.

In this problem context:

• You are akin to a collector trying to gather each letter in "TIDE" from random draws in the detergent packets.


• The challenge lies in obtaining one complete set and determining how these collections can be achieved without forming additional complete sets by chance.


• This kind of problem is usually analyzed by understanding the probability distribution of the waiting time until collecting all needed items (letters, in this case), and then analyzing how to prevent any further complete sets within the additional available attempts.

The key element of interest is the need to avoid a second full set while ensuring every letter appears at least once in a set of draws, reflecting the balancing act required in the collector's problem.