Modern Math

Permutation & Combination

Medium

The number of ways in which coins of denominations of rupees 2, 5 and, 10 can be combined to make a value of rupees 50 is

✅ Correct Option: 2

Let $a$ , $b$ , $c$ be the number of Rs $2$ , Rs $5$ , and Rs $10$ coins.

$2a + 5b + 10c = 50$

Fix $c$ and count valid $(a, b)$ pairs. The equation becomes:

$2a + 5b = 50 - 10c$

The right side is even, so $5b$ must be even, meaning $b$ must be even.

For each value of $c$ , list even values of $b$ such that $5b \leq 50 - 10c$ :

$c = 0$ : $b \in \{0, 2, 4, 6, 8, 10\}$ gives $6$ ways

$c = 1$ : $b \in \{0, 2, 4, 6, 8\}$ gives $5$ ways

$c = 2$ : $b \in \{0, 2, 4, 6\}$ gives $4$ ways

$c = 3$ : $b \in \{0, 2, 4\}$ gives $3$ ways

$c = 4$ : $b \in \{0, 2\}$ gives $2$ ways

$c = 5$ : $b = 0$ gives $1$ way

Total $= 6 + 5 + 4 + 3 + 2 + 1 = 21$

Answer $= 21$

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