STATEMENT-1 :The total number of combinations of n thing by taking some or all is `2^(n)`.
STATEMENT -2 : 5 identical balls can be distributed among 10 identical boxes in only one way if not more than one ball can go into a box.
STATEMENT-3 : In the permutations of n things taken r at a time , the number of permutations in which m particular things occur together is `""^(n - m)P_(r - m) * ""^(r)P_(m)`

Options

A. (a)F T F
B. (b)F F F
C. (c)T F T
D. (d)F T T

Answer

To solve the problem, we will analyze each statement one by one.

### Step 1: Analyze Statement 1
**Statement 1**: The total number of combinations of n things by taking some or all is \(2^n\).

To understand this statement, we can use the concept of combinations. The number of ways to choose \(k\) items from \(n\) items is given by \(C(n, k)\). The total number of combinations when taking some or all items is the sum of combinations for all possible values of \(k\) from 0 to \(n\):

\[ C(n, 0) + C(n, 1) + C(n, 2) + \ldots + C(n, n) = 2^n \]

This is derived from the binomial theorem, which states that:

\[ (1 + 1)^n = \sum_{k=0}^{n} C(n, k) = 2^n \]

Thus, **Statement 1 is true**.

### Step 2: Analyze Statement 2
**Statement 2**: 5 identical balls can be distributed among 10 identical boxes in only one way if not more than one ball can go into a box.

Since the balls are identical and we cannot place more than one ball in a box, the only way to distribute 5 identical balls into 10 identical boxes is to place one ball in each of 5 boxes, leaving the other 5 boxes empty.

Thus, there is only **one way** to do this. Therefore, **Statement 2 is true**.

### Step 3: Analyze Statement 3
**Statement 3**: In the permutations of n things taken r at a time, the number of permutations in which m particular things occur together is given by:

\[ (n - m)P_{(r - m)} \times rP_m \]

To understand this, we can treat the m particular things as a single unit or "block". This reduces the problem to finding the permutations of \( (n - m + 1) \) items (the block plus the remaining \( n - m \) items).

The number of ways to arrange this block with the remaining items is:

\[ (n - m + 1)P_r \]

Within the block, the m items can be arranged among themselves in \( m! \) ways. Thus, the total number of arrangements is:

\[ (n - m + 1)P_{(r - m)} \times m! \]

This matches the statement given, so **Statement 3 is true**.

### Conclusion
- Statement 1: True
- Statement 2: True
- Statement 3: True

Thus, all statements are true.

### Final Answer
The correct option is **D** (all statements are true).