On a target board with scores 0, 1, 3, and 5, is it possible to obtain a score of 12 or 14 with exactly three coins?

On a target board with scores 0, 1, 3, and 5, is it possible to obtain a score of 12 or 14 with exactly three coins?

• A. No; these scores are impossible.
• B. Yes; these scores are possible.
• C. Only 12 is possible.
• D. Only 14 is possible.

Answer: (A)

You’re testing how sums form when you pick three values from 0, 1, 3, and 5. The key idea is to think about how many nonzero picks you use, since 0 is even and 1, 3, 5 are odd. - If none are nonzero, the total is 0. - If exactly one is nonzero, the total is one of 1, 3, or 5 (since you’re just picking a single odd number). - If exactly two are nonzero, you’re adding two odd numbers, which always gives an even total. The possible sums are 1+1=2, 1+3=4, 1+5=6, 3+3=6, 3+5=8, and 5+5=10. - If all three are nonzero, you’re adding three odd numbers, which is odd. The possible totals include 1+1+1=3 up to 5+5+5=15, giving values like 3, 5, 7, 9, 11, 13, 15. Putting those together, the totals you can achieve with exactly three coins are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, and 15. The numbers 12 and 14 do not appear in this list, so they cannot be obtained with exactly three coins.

You’re testing how sums form when you pick three values from 0, 1, 3, and 5. The key idea is to think about how many nonzero picks you use, since 0 is even and 1, 3, 5 are odd.

• If none are nonzero, the total is 0.

• If exactly one is nonzero, the total is one of 1, 3, or 5 (since you’re just picking a single odd number).

• If exactly two are nonzero, you’re adding two odd numbers, which always gives an even total. The possible sums are 1+1=2, 1+3=4, 1+5=6, 3+3=6, 3+5=8, and 5+5=10.

• If all three are nonzero, you’re adding three odd numbers, which is odd. The possible totals include 1+1+1=3 up to 5+5+5=15, giving values like 3, 5, 7, 9, 11, 13, 15.

Putting those together, the totals you can achieve with exactly three coins are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, and 15. The numbers 12 and 14 do not appear in this list, so they cannot be obtained with exactly three coins.