The number of ways in which $4$ different things can be distributed to $6$ persons so that no person gets all the things is

  • A
    $1292$
  • B
    $1296$
  • C
    $1290$
  • D
    $4090$

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Consider the following statements:
$(i)$ The number of one-one functions from set $A$ to set $B$,where $O(A) = m$ and $O(B) = n$ $(m \leq n)$,is given by ${}^n P_m$.
(ii) The number of ways in which $n$ people can be arranged at a circular table is $\frac{(n-1)!}{2}$.
(iii) The number of ways of selecting at least one thing out of the given $n$ distinct things is $2^n - 1$.
(iv) The number of ways in which $n$ distinguishable objects can be distributed into $k$ distinguishable bins is ${}^n C_{k-1}$.
Which of the following is true?

The sum of all the four-digit numbers that can be formed using all the digits $2, 1, 2, 3$ is equal to $.......$.

If $S_n = \sum_{r=0}^n \frac{1}{^nC_r}$ and $T_n = \sum_{r=0}^n \frac{r}{^nC_r}$,then $\frac{T_n}{S_n}$ is equal to

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The number of even numbers greater than $1000000$ that can be formed using all the digits $1, 2, 0, 2, 4, 2, 4$ is

An eight-digit number divisible by $9$ is to be formed using digits from $0$ to $9$ without repeating the digits. The number of ways in which this can be done is: (in $(7!)$)

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