Question
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
The problem asks for the number of subsets B of the set $S = \{1, 2, ..., 11\}$ such that the size of the subset, denoted as $n(B)$, is greater than or equal to 2 ($n(B) \geq 2$), and the product of all elements in B is an even number.
Set S and Subset Properties
- The given set is $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$. The total number of elements in S is $n(S) = 11$.
- The total number of possible subsets of S is $2^{n(S)} = 2^{11} = 2048$.
- A product of numbers is even if at least one of the numbers is even.
- Conversely, a product of numbers is odd only if all the numbers are odd.
Calculating Subsets with Odd Product
- First, let's find the subsets where the product of elements is odd. This happens only when all elements in the subset are odd numbers.
- Identify the odd numbers in S: $O = \{1, 3, 5, 7, 9, 11\}$. The count of odd numbers is $n(O) = 6$.
- The subsets containing only odd numbers are the subsets of O. The number of such subsets is $2^{n(O)} = 2^6 = 64$.
Calculating Subsets with Even Product
- The number of subsets where the product of elements is even is the total number of subsets minus the number of subsets with an odd product.
- Number of subsets with even product = Total subsets - Number of subsets with odd product
- Number of subsets with even product = $2^{11} - 2^6 = 2048 - 64 = 1984$.
Applying the Size Condition $n(B) \geq 2$
- We need subsets with an even product AND $n(B) \geq 2$.
- The 1984 subsets with an even product include subsets of all sizes (0, 1, 2, ...).
- We need to exclude subsets from these 1984 that have $n(B) < 2$. These are subsets with size 0 or 1.
- Case 1: $n(B) = 0$ (Empty Set): The empty set $\emptyset$ has a product conventionally defined as 1 (odd). It's not included in the 1984 count.
- Case 2: $n(B) = 1$ (Singleton Sets):
- Subsets with one odd element: $\{1\}, \{3\}, \{5\}, \{7\}, \{9\}, \{11\}$. Their product is odd. These 6 subsets are part of the 64 subsets with an odd product and are *not* included in the 1984 count.
- Subsets with one even element: $\{2\}, \{4\}, \{6\}, \{8\}, \{10\}$. Their product is even. There are 5 such subsets. These 5 subsets *are* included in the 1984 count.
- Since the condition is $n(B) \geq 2$, we must remove the 5 singleton subsets with even elements from the count of 1984.
- The number of elements in set A = (Number of subsets with even product) - (Number of singleton subsets with even product)
- Number of elements in A = $1984 - 5 = 1979$.
Final Result
The number of subsets B of S such that $n(B) \geq 2$ and the product of elements in B is even is 1979.
Was this answer helpful?
Similar Questions
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
- Let $a_1 = 1$ and for $n \ge 1, a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\left| \sum_{n=1}^\infty \left( a_n - \frac{2}{n^2} \right) \right|$ is equal to ____________.
If $ \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)$ $\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\right)$ $\dots$ $ \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right)$ = $\frac{\alpha^{13}}{^{14}C_0 ^{14}C_1 \dots ^{14}C_{12}} $, then $30\alpha$ is equal to __________.
- Three persons enter in a lift at the ground floor. The lift will go upto 10$^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to _________.
- The number of 4-letter words, with or without meaning, which can be formed using the letters PQR PQRSTUVP, is ___________.
- Let $|A| = 6$, where A is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n$, $m, n \in \mathbf{N}$, then $m + n$ is equal to ___________.
- Let $A = \{2, 3, 4, 5, 6\}$. Let $R$ be a relation on the set $A \times A$ given by $(x, y) R (z, w)$ if and only if $x$ divides $z$ and $y \le w$. Then the number of elements in $R$ is __________.
Important Questions from Algebra
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
- Let $a_1 = 1$ and for $n \ge 1, a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\left| \sum_{n=1}^\infty \left( a_n - \frac{2}{n^2} \right) \right|$ is equal to ____________.
Need Expert Advice?
More Questions from JEE Main
