Give step-by-step solution with explanation and final answer:Tourists returning home were interviewed about which countries they had visited
on their trip abroad. Of the 265 tourists interviewed, all

Question

Give step-by-step solution with explanation and final answer:Tourists returning home were interviewed about which countries they had visited
on their trip abroad. Of the 265 tourists interviewed, all of them had visited at
least one of Germany, Nepal, or Vietnam, while exactly 10 had visited all three.
150 tourists had travelled to Vietnam and 1 15 had visited Germany.
Additionally, 150 of them had not been to Nepal.

The interviews also revealed that 60 tourists had been to Vietnam only. If we also
know that 55 had travelled to Germany and Vietnam but not to Nepal, then how
many people had visited Nepal only?

025

O35

60

(O65

0125

VIPSolutions · Accepted Answer

Let's define:

- $ G $: Visited Germany
- $ N $: Visited Nepal
- $ V $: Visited Vietnam

Let:
- $ n = |N| $: Total who visited Nepal
- $ g = |G| = 115 $: Total who visited Germany
- $ v = |V| = 150 $: Total who visited Vietnam
- $ |N \cap G \cap V| = 10 $: Visited all three
- $ |V| = 150 $
- $ |G| = 115 $
- $ |N'| = 150 $: Did **not** visit Nepal (so visited Germany or Vietnam or both, but not Nepal)
- $ |V \text{ only}| = 60 $
- $ |G \cap V \cap N'| = 55 $: Germany and Vietnam but not Nepal

We are to find **the number who visited Nepal only**.

### Step 1: Use the Principle of Inclusion-Exclusion (PIE)

Total tourists = $ 265 $

Let $ x $ = number who visited Nepal only.

Let’s define:
- $ A = $ Nepal only
- $ B = $ Germany only
- $ C = $ Vietnam only ($ C = 60 $)
- $ D = $ Nepal and Germany only
- $ E = $ Nepal and Vietnam only
- $ F = $ Germany and Vietnam only ($ F = 55 $)
- $ G = $ All three ($ G = 10 $)

So:

$$
265 = x + B + 60 + D + E + 55 + 10
$$
$$
265 = x + B + D + E + 125
$$
$$
x + B + D + E = 140 \tag{1}
$$

### Step 2: Use info about "not been to Nepal" ($ |N'| = 150 $)

People who **did not visit Nepal**:
- Germany only ($ B $)
- Vietnam only ($ 60 $)
- Germany and Vietnam only ($ 55 $)

So:

$$
B + 60 + 55 = 150
$$
$$
B = 35
$$

### Step 3: Use totals for Germany and Vietnam

**Visited Germany** ($ 115 $):
- Germany only ($ B = 35 $)
- Germany and Nepal only ($ D $)
- Germany and Vietnam only ($ 55 $)
- All three ($ 10 $)

So:
$$
B + D + 55 + 10 = 115
$$
$$
35 + D + 65 = 115
$$
$$
D = 115 - 100 = 15
$$

**Visited Vietnam** ($ 150 $):
- Vietnam only ($ 60 $)
- Nepal and Vietnam only ($ E $)
- Germany and Vietnam only ($ 55 $)
- All three ($ 10 $)

So:
$$
60 + E + 55 + 10 = 150
$$
$$
E = 150 - 125 = 25
$$

### Step 4: Substitute values back to (1):

Recall $ x + B + D + E = 140 $  
We have: $ B = 35 $, $ D = 15 $, $ E = 25 $

$$
x + 35 + 15 + 25 = 140
$$
$$
x + 75 = 140
$$
$$
x = 65
$$

---

## **Final Answer**

$$
\boxed{65}
$$

**65 people had visited Nepal only.**

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