Mental math: is there a shortcut to solve this mentally Q:  A bank contains pennies, nickels, dimes, quarters and half-dollars. <br>How many different sets of three coins can be...

Mental math: is there a shortcut to solve this mentally

Q:  A bank contains pennies, nickels, dimes, quarters and half-dollars. <br>How many different sets of three coins can be formed?______

A:35

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I'm sorry, I worked the problem as a permutation rather than a combination.

The formula for a combination when repetition is allowed:

(n + r - 1)!
-------------
r!(n - 1)!

where n is the possible choices and r is the number of choices that are selected.

In this case, n would be 5 because there are 5 different types of coins and r would be 3 because you are looking for combinations of 3 coins.

Substitute 5 in for n and 3 in for r.  Then add & subtract to simplify.

(n + r - 1)!
-------------
r!(n - 1)!

(5 + 3 - 1)!
-------------
3!(5 - 1)!

7!
-------
3! * 4!

Here is where the mental math will come in handy.

7 * 6 * 5 * 4 * 3 * 2 * 1
----------------------------
3 * 2 * 1 * 4 * 3 * 2 * 1

Simplify by crossing out any numbers that appear in both the numerator and the denominator.  In this case, 4 * 3 * 2 * 1 can be eliminated from the fraction.

7 * 6 * 5
----------
3 * 2 * 1

Since 3 * 2 * 1 = 6 and 6 is also in the numerator, 6 can be eliminated.

7 * 5 = 35

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