Problem 149 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Column-I (I) The sum of the digits in the unit's place of all the numbers formed with the help of \(3,4,5\), 6 taken all at a time is... (II) The number of ways in which a committee of 5 can be chosen from 10 candidates so as to exclude the youngest if it includes the oldest, is... (III) The number of divisors of 9600 including 1 and 9600 are... (IV) The number of ways of choosing 10 balls from infinite white, red, blue and green balls is... Column-II (A) 286 (B) 108 (C) 196 (D) 48

Step-by-Step Solution

Answer

I (108) B, II (70) - not listed, III (48) D, IV (286) A.

Step 1: Calculate Units Digit Sum for Numbers Formed by 3, 4, 5, 6

The numbers formed using the digits 3, 4, 5, and 6 are permutations of these four digits. Each digit will appear at every position (including the unit's place) equally as often. There are 4! = 24 total permutations. Since each of the four digits will appear in the unit's place in exactly 6 of the permutations (24 permutations / 4 digits), we calculate the sum of these digits: \(3 + 4 + 5 + 6 = 18\). Therefore, the total sum of the unit's place digits is \(6 \times 18 = 108\).

Step 2: Find Combinations of Committee Excluding Youngest with Oldest Included

To form a committee of 5 from 10 candidates excluding the youngest if the oldest is included, first fix the oldest member. This leaves us choosing 4 more members from the remaining 8 candidates (since we exclude the youngest from the remaining 9). The number of ways to choose 4 members from 8 is given by the combination formula \( \binom{8}{4} \). Calculating this gives \( \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70\).

Step 3: Count the Divisors of 9600

The prime factorization of 9600 is \(9600 = 2^7 \times 3^1 \times 5^2\). To find the total number of divisors, use the formula \((a+1)(b+1)...\) where \(a, b, ...\) are the exponents in the factorization: \((7+1)(1+1)(2+1) = 8 \times 2 \times 3 = 48\). Thus, there are 48 divisors.

Step 4: Determine Ways to Choose 10 Balls from 4 Colors

We have an infinite number of each color (white, red, blue, green) and need to choose 10 balls total. This is a classical "stars and bars" problem where we distribute 10 identical items into 4 distinct categories. The formula for this is \( \binom{n+k-1}{k-1} \) where \( n \) is the total number of items (balls) and \( k \) is the number of categories (colors): \( \binom{10+4-1}{4-1} = \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286\).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focusing on counting, arrangements, and combinations. It includes techniques that allow us to handle different types of counting problems. These problems usually don't involve large calculations but require careful reasoning orchestrated by combinatorial principles.

• Permutations: These are arrangements of a set of items. If you have a set of "n" items and you want to arrange "r" of them at a time, the permutations are calculated using the formula \( P(n, r) = \frac{n!}{(n-r)!} \).
• Combinations: Unlike permutations, combinations refer to the selection of items from a set without regard to the order. The formula for combinations is \( C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

Combinatorics helps solve problems like forming a committee or distributing items into different groups, as seen in many mathematical exercises. Understanding how to count effectively with these tools is key to solving combinatorial problems.

Divisors

Divisors are numbers that divide another number completely without leaving a remainder. Understanding divisors is essential in number theory and many mathematical problems related to factors.
To find the number of divisors of a given number, we first perform its prime factorization. Any integer can be expressed as a product of prime numbers with specific powers.
For example, let's consider 9600. The prime factorization of 9600 is \(2^7 \times 3^1 \times 5^2\). To find the total number of divisors, we apply the formula \( (a+1)(b+1)... \), where "a", "b", etc., are the exponents of the prime factors.

• For 9600, it becomes: \( (7+1)(1+1)(2+1) = 8 \times 2 \times 3 = 48\).

Thus, 9600 has 48 divisors. This prime factorization method can be applied to any number to efficiently determine its total number of divisors.

Stars and Bars Method

The stars and bars method is a combinatorial technique used to distribute identical items into distinct categories. It's ideal for problems where the order among items in sets isn't important.
Imagine you have several identical balls that need to be placed into different colored bins (like white, red, blue, and green). The challenge is to calculate how many ways you can do this given unlimited items for each color.
Here's how it works:

• "Stars" represent the items you are distributing.
• "Bars" are the dividers placed between different categories.

The formula for stars and bars is \( \binom{n+k-1}{k-1} \), where *n* is the number of items and *k* is the number of categories. For instance, if selecting 10 balls from 4 available colors, calculate as:
\[ \binom{10+4-1}{4-1} = \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \]
This result provides the number of solutions to how you can distribute the balls among the bins.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks—prime numbers. Understanding this concept is fundamental in various areas of mathematics, especially in finding divisors.
To achieve prime factorization, continuously divide the number by the smallest prime number until each resulting factor is prime:

• Start with the smallest prime numbers: Begin with 2, the smallest prime. If the number is even, divide by 2 repeatedly until it's odd.
• Move to the next smallest primes: After 2, proceed to 3, 5, 7, and so forth, dividing the number completely by these primes.

Consider breaking down the number 9600. By systematic division:\[9600 = 2^7 \times 3^1 \times 5^2\]Achieving this expression helps compute how many divisors a number has and is essential for simplifying expressions, solving equations, and other mathematical applications.