How many arithmetic progressions with $10$ terms are there, whose first term is in the set ${1, 2, 3, 4}$ and whose common difference is in the set ${3, 4, 5, 6, 7}$ ?

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Solution

The first term can be any of the three numbers from the set $1, 2, 3 & 4$ . Therefore, the first term can be chosen in $4$ ways. The common difference can be chosen in $5$ ways. Hence total number of ways $= 4 \times 5 = 20$

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