Question

penny has 25 dimes. she likes to arrange them into three piles, putting an odd number of dimes into each pile. in how many ways could she do this?

Answer 1

10

Answer 2

78

Answer 3

7 ways

Answer 4

i think it is 78 too

Answer 5

Answer = 75
Explanation:-
As information provided in the question,

          Total number of dimes that Penny has = 25 dimes

          Total number of piles in which she likes to arrange the 25 dimes = 3 piles

Now,

      It is given that she wants to put an odd number of dimes into each pile.

Thus,

    Total number of possible pairs of three odd numbers that has a sum of 25 are as:

        {1, 3, 21}, {1, 5, 19}, {1, 7, 17}, {1, 9, 15}, {1, 11, 13}

        {3, 3, 19}, {3, 5, 17}, {3, 7, 15}, {3, 9, 13}, {3, 11, 11}

        {5, 5, 15}, {5, 7, 13}, {5, 9, 11}

        {7, 7, 11}, {7, 9, 9}

Now,

      There are total 15 pairs in which 10 pairs have all three difinite elements and 5 pairs have two identical elements in each pair.

Therefore,

       The number of ways in which we can arrange each pair which has all three distinct elements = 6 ways

And, The number of ways in which we can arrange each pair which has two identical elements elements = 3 ways 

Therefore,

      Total number of ways in which Penny can arrange all 25 dimes into three piles = 10 x 6 + 5 x 3

                                                                                                                                   = 60 + 15

                                                                                                                                   = 75 ways