Home
Class 11
MATHS
0!=0...

0!=0

Text Solution

AI Generated Solution

The correct Answer is:
To determine whether the statement "0! = 0" is true or false, we need to understand the definition of factorial, particularly for the case of zero. ### Step-by-Step Solution: 1. **Understanding Factorial**: The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers up to \( n \). For example: - \( 3! = 3 \times 2 \times 1 = 6 \) - \( 2! = 2 \times 1 = 2 \) - \( 1! = 1 \) 2. **Defining Zero Factorial**: By definition, the factorial of zero, \( 0! \), is defined to be equal to 1. This is a convention in mathematics that helps maintain consistency in various formulas, particularly in combinatorics. 3. **Mathematical Justification**: The reason \( 0! = 1 \) can be understood through the concept of permutations. The number of ways to arrange zero objects is exactly one way: doing nothing. Therefore, \( 0! = 1 \). 4. **Conclusion**: Since \( 0! = 1 \), we can conclude that the statement "0! = 0" is false. ### Final Answer: The statement "0! = 0" is **false** because \( 0! = 1 \). ---
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • PERMUTATIONS AND COMBINATIONS

    CBSE COMPLEMENTARY MATERIAL|Exercise SECTION-B(Short anwer type question)|11 Videos

  • PERMUTATIONS AND COMBINATIONS

    CBSE COMPLEMENTARY MATERIAL|Exercise Section - C(Short answer type question)|21 Videos

  • PERMUTATIONS AND COMBINATIONS

    CBSE COMPLEMENTARY MATERIAL|Exercise SECTION-A(MCQ)|10 Videos

  • MATHEMATICAL REASONING

    CBSE COMPLEMENTARY MATERIAL|Exercise SECTION-B SHORT ANSWER TYPE QUESTIONS|5 Videos

  • PRACTICE PAPER DESIGN - I

    CBSE COMPLEMENTARY MATERIAL|Exercise SECTION -D|6 Videos

Similar Questions

Explore conceptually related problems

Choose the correct answer in questions 17 to 19: If x, y, z are nonzero real numbers then the inverse of metrix A=[{:(x,0,0),(0,y,0),(0,0,z):}] is : (a) [{:(x^(-1),0,0),(0,y^(1),0),(0,0,z^(1)):}] (b) xyz[{:(x^(-1),0,0),(0,y^(1),0),(0,0,z^(1)):}] ( c) 1/(xyz)[{:(x,0,0),(0,y,0),(0,0,z):}] (d) 1/(xyz)[{:(1,0,0),(0,1,0),(0,0,1):}]

" if " f(0) = |{:(sin 0 ,,cos 0,,sin 0),(cos0,,sin0,,cos0),(cos0,, sin 0,,sin 0):}| then

Knowledge Check

  • let P_(1)=[(1,0,0),(0,1,0),(0,0,1)],P_(2)=[(1,0,0),(0,0,1),(0,1,0)],P_(3)=[(0,1,0),(1,0,0),(0,0,1)],P_(4)=[(0,1,0),(0,0,1),(1,0,0)],P_(5)=[(0,0,1),(1,0,0),(0,1,0)],P_(6)=[(0,0,1),(0,1,0),(1,0,0)] and X=sum_(k=1)^(6) P_(k)[(2,1,3),(1,0,2),(3,2,1)]P_(k)^(T) Where P_(k)^(T) is transpose of matrix P_(k) . Then which of the following options is/are correct?

    A
    X is a symmetric matrix
    B
    if `X=[(1),(1),(1)]=alpha[(1),(1),(1)]`, then `alpha=30`
    C
    X-30I is an invertible matrix
    D
    The sum of diagonal entries of X is 18.

  • Let P_1=I=[{:(1,,0,,0),(0,,1,,0),(0,,0,,1):}],P_2=[{:(1,,0,,0),(0,,0,,1),(0,,1,,0):}] P_3=[{:(0,,1,,0),(1,,0,,0),(0,,0,,1):}],P_4=[{:(0,,1,,0),(0,,0,,1),(1,,0,,0):}] P_5=[{:(0,,0,,1),(1,,0,,0),(0,,1,,0):}],P_6=[{:(0,,0,,1),(0,,1,,0),(1,,0,,0):}] and X=sum_(k=1)^(6)P_(K)[{:(2,,1,,3),(1,,0,,2),(3,,2,,1):}]P_(K)^(T) Where , P_(K)^(T) denotes the transpose of the matrix P_(X) . then which of the following option is/are correct ?

    A
    X is a symmertic matrix
    B
    The sum of diagonal entries of X is 18
    C
    `X-30 I` is an invertible matrix
    D
    If`X[{:(1),(1),(1):}]=alpha[{:(1),(1),(1):}],then" "alpha=30`

  • Let P_1=I=[(1,0,0),(0,1,0),(0,0,1)], P_2=[(1,0,0),(0,0,1),(0,1,0)], P_3=[(0,1,0),(1,0,0),(0,0,1)] , P_4=[(0,1,0),(0,0,1),(1,0,0)], P_5=[(0,0,1),(1,0,0),(0,1,0)], P_6=[(0,0,1),(0,1,0),(1,0,0)] and X=sum_(k=1)^6[(2,1,3),(1,0,2),(3,2,1)]P_k^T where P_k^T denotes the transpose of the matrix P_k . Then which of the following options is(are) correct ?

    A
    The sum of diagonal entries of X is 18
    B
    X is a symmetric matrix
    C
    X=30I is an invertible matrix
    D
    If `X[(1),(1),(1)]=alpha[(1),(1),(1)]`, then `alpha`=30
    • Home
    Profile