To solve the problem, we need to find a decimal number between 0.36 and 0.42 using three different digits. Let's break down the steps.
1. Identify the Range:
- The lower bound is [tex]\(\frac{36}{100} = 0.36\)[/tex].
- The upper bound is [tex]\(\frac{42}{100} = 0.42\)[/tex].
2. Three-Digit Decimal:
- The number Alice needs to form is a three-digit decimal in the form [tex]\(0.abc\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are digits from 0 to 9 and none of them are the same.
3. Determine Valid Digits:
- We need to find all combinations of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] such that [tex]\(0.36 < 0.abc < 0.42\)[/tex].
4. Forming and Validating Numbers:
The following are the valid three-digit decimals that Alice could form:
- 0.360
- 0.361
- 0.362
- 0.364
- 0.365
- 0.367
- 0.368
- 0.369
- 0.370
- 0.371
- 0.372
- 0.374
- 0.375
- 0.376
- 0.378
- 0.379
- 0.380
- 0.381
- 0.382
- 0.384
- 0.385
- 0.386
- 0.387
- 0.389
- 0.390
- 0.391
- 0.392
- 0.394
- 0.395
- 0.396
- 0.397
- 0.398
- 0.401
- 0.402
- 0.403
- 0.405
- 0.406
- 0.407
- 0.408
- 0.409
- 0.410
- 0.412
- 0.413
- 0.415
- 0.416
- 0.417
- 0.418
- 0.419
Each of these numbers satisfies the requirement of being between 0.36 and 0.42. Thus, Alice has numerous possible combinations of digits to choose from to form a valid decimal number within the specified range.
