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# help counting

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In how many ways can you change a 20 dollar bill using 10 dollar bills, 5 dollar bills, 2 dollar bills, and 1 dollar bills?

Jun 10, 2021

#1
+505
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Now in this problem,

Let the no. of  10 dollar bills be x, 5 dollar bills be y, 2 dollar bills be z and 1 dollar bills be w.

According to question,

$$10x+5y+2z+w=20$$                    $$...(1)$$

In order to find no. of ways, we've to find no. of solutions eq. (1) has.

∴ By the theory of combinatrics,

No. of solutions $$=\left( \begin{array}{c} 20+4-1 \\ 4-1 \end{array} \right)$$

$$=\left( \begin{array}{c} 23 \\ 3 \end{array} \right)$$

$$=1771$$

∴ Eq. (1) has 1771 possible set of solutions.

Thus a \$20 bill can be changed in 1771  possible ways.

~Amy

Jun 10, 2021
edited by amygdaleon305  Jun 10, 2021
#2
+174
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Each ball has five options to go

Therefore four balls  will be filled in =

5×5×5×5

= 5^4 ways = 625 ways

Jun 10, 2021
#3
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10 ca we changed into 8 different ways:-

10

5,5

5,2,2,1

2,2,2,2,1,1

2,2,2,1,1,1,1

2,2,1,1,1,1,1,1

2,1,1,1,1,1,1,1,1

8 1s

Since 10,10 counts as 1

so we know that there will be a total of 8+7 different ways to change