Q.4. The number of functions , which are not onto, is :

Q: What is the correct answer for JEE Main Mathematics Q4?

The correct answer is B. Sol. The total number of functions from a set of 4 elements to a set of 3 elements is $3^4 = 81$. The number of onto functions from a set of $m$ elements to a set of $n$ elements is given by $n! \times S(m, n)$, where $S(m, n)$ is the Stirling number of the second kind. Alternatively, using the Principle of Inclusion-Exclusion, the number of onto functions is $3^4 - \binom{3}{1}(3-1)^4 + \binom{3}

Q: Q.4. The number of functions , which are not onto, is :

The correct answer is B. Sol. The total number of functions from a set of 4 elements to a set of 3 elements is $3^4 = 81$. The number of onto functions from a set of $m$ elements to a set of $n$ elements is given by $n! \times S(m, n)$, where $S(m, n)$ is the Stirling number of the second kind. Alternatively, using the Principle of Inclusion-Exclusion, the number of onto functions is $3^4 - \binom{3}{1}(3-1)^4 + \binom{3}

Q: What are the options for this JEE Main Mathematics question?

A. 48 · B. 45 · C. 51 · D. 35

Question

Q.4. The number of functions , which are not onto, is :

FirstInTest · Accepted Answer

Sol. The total number of functions from a set of 4 elements to a set of 3 elements is $3^4 = 81$. The number of onto functions from a set of $m$ elements to a set of $n$ elements is given by $n! 	imes S(m, n)$, where $S(m, n)$ is the Stirling number of the second kind. Alternatively, using the Principle of Inclusion-Exclusion, the number of onto functions is $3^4 - \binom{3}{1}(3-1)^4 + \binom{3}{2}(3-2)^4 - \binom{3}{3}(3-3)^4 = 3^4 - 3 \cdot 2^4 + 3 \cdot 1^4 - 0 = 81 - 3 \cdot 16 + 3 \cdot 1 = 81 - 48 + 3 = 36$. The number of functions that are NOT onto is Total functions - Onto functions = $3^4 - 36 = 81 - 36 = 45$. The solution provided uses the formula for 'No. of into function' which is $^nC_r (r^m - ^rC_1 (r-1)^m + ...)$. Here, it seems to be $^3C_2 (2^4 - ^2C_1 1^4) = 3(16 - 2) = 3 	imes 14 = 42$. This is incorrect. The formula for number of onto functions from a set of size $m$ to a set of size $n$ is $\sum_{k=0}^{n} (-1)^k \binom{n}{k} (n-k)^m$. For $m=4, n=3$: $\binom{3}{0}(3-0)^4 - \binom{3}{1}(3-1)^4 + \binom{3}{2}(3-2)^4 - \binom{3}{3}(3-3)^4 = 1 \cdot 3^4 - 3 \cdot 2^4 + 3 \cdot 1^4 - 1 \cdot 0^4 = 81 - 3 \cdot 16 + 3 \cdot 1 = 81 - 48 + 3 = 36$. Number of functions which are not onto = Total functions - Number of onto functions = $3^4 - 36 = 81 - 36 = 45$. The solution in the image calculates 'No. of into function' as $^3C_2 2^4 - ^3C_1 1^4$. This is a specific case of the inclusion-exclusion principle for $n=3$, which is $3^4 - \binom{3}{1}2^4 + \binom{3}{2}1^4$. The image's calculation is $3 	imes 16 - 3 	imes 1 = 48 - 3 = 45$. This is the correct calculation for non-onto functions.

Answer

48

Answer

45

Answer

51

Answer

35

Related Questionsin Mathematics

More from JEE Main

Browse Engineering Questions by Subject

Practice More JEE Main Questions