To solve the problem step by step, we will break it down into parts as per the requirements of the question.
### Step 1: Calculate the total number of ways to give none or some coins to a beggar.
The person has:
- 3 coins of 25 paise
- 4 coins of 50 paise
- 2 coins of 1 rupee
For each type of coin, the person can choose to give 0 to the maximum number of coins he has.
1. **Coins of 25 paise**: He can give 0, 1, 2, or 3 coins. This gives us 4 options (0 to 3).
2. **Coins of 50 paise**: He can give 0, 1, 2, 3, or 4 coins. This gives us 5 options (0 to 4).
3. **Coins of 1 rupee**: He can give 0, 1, or 2 coins. This gives us 3 options (0 to 2).
Now, we multiply the number of options for each type of coin:
\[
\text{Total ways} = (3 + 1)(4 + 1)(2 + 1) = 4 \times 5 \times 3
\]
Calculating this gives:
\[
4 \times 5 = 20
\]
\[
20 \times 3 = 60
\]
Thus, the total number of ways he can give none or some coins to a beggar is **60**.
### Step 2: Calculate the number of ways to give at least one coin of 1 rupee.
When he must give at least one coin of 1 rupee, we adjust the options:
1. **Coins of 25 paise**: Still 4 options (0 to 3).
2. **Coins of 50 paise**: Still 5 options (0 to 4).
3. **Coins of 1 rupee**: Now he can give 1 or 2 coins. This gives us 2 options (1 to 2).
Now, we multiply the number of options again:
\[
\text{Total ways} = (3 + 1)(4 + 1)(2) = 4 \times 5 \times 2
\]
Calculating this gives:
\[
4 \times 5 = 20
\]
\[
20 \times 2 = 40
\]
Thus, the number of ways he can give at least one coin of 1 rupee is **40**.
### Step 3: Calculate the number of ways to give at least one coin of each kind.
For this case, he must give at least one coin of each type:
1. **Coins of 25 paise**: He can give 1, 2, or 3 coins. This gives us 3 options (1 to 3).
2. **Coins of 50 paise**: He can give 1, 2, 3, or 4 coins. This gives us 4 options (1 to 4).
3. **Coins of 1 rupee**: He can give 1 or 2 coins. This gives us 2 options (1 to 2).
Now, we multiply the number of options:
\[
\text{Total ways} = (3)(4)(2) = 3 \times 4 \times 2
\]
Calculating this gives:
\[
3 \times 4 = 12
\]
\[
12 \times 2 = 24
\]
Thus, the number of ways he can give at least one coin of each kind is **24**.
### Summary of Results:
- Total ways to give none or some coins: **60**
- Ways to give at least one coin of 1 rupee: **40**
- Ways to give at least one coin of each kind: **24**
