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# Probability Problem | Combinatorics | AIME I, 2015 - Question 5

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 based on Probability.

## Probability Problem - AIME I, 2015

In a drawer Sandy has 5 pairs of socks, each pair a different color. on monday sandy selects two individual socks at random from the 10 socks in the drawer. On tuesday Sandy selects 2 of the remaining 8 socks at random and on wednesday two of the remaining 6 socks at random. The probability that wednesday is the first day Sandy selects matching socks is $\frac{m}{n}$, where m and n are relatively prime positive integers, find m+n.

• is 107
• is 341
• is 840
• cannot be determined from the given information

### Key Concepts

Algebra

Theory of Equations

Probability

## Check the Answer

AIME, 2015, Question 5

Geometry Revisited by Coxeter

## Try with Hints

First hint

Wednesday case - with restriction , select the pair on wednesday in $5 \choose 1$ ways

Tuesday case - four pair of socks out of which a pair on tuesday where a pair is not allowed where 4 pairs are left,the number of ways in which this can be done is $8 \choose 2$ - 4

Second Hint

Monday case - a total of 6 socks and a pair not picked $6 \choose 2$ -2

Final Step

by multiplication and principle of combinatorics $\frac{(5)({5\choose 2} -4)({6 \choose 2}-2)}{{10 \choose 2}{8 \choose 2}{6 \choose 2}}$=$\frac{26}{315}$. That is 341.

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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 based on Probability.

## Probability Problem - AIME I, 2015

In a drawer Sandy has 5 pairs of socks, each pair a different color. on monday sandy selects two individual socks at random from the 10 socks in the drawer. On tuesday Sandy selects 2 of the remaining 8 socks at random and on wednesday two of the remaining 6 socks at random. The probability that wednesday is the first day Sandy selects matching socks is $\frac{m}{n}$, where m and n are relatively prime positive integers, find m+n.

• is 107
• is 341
• is 840
• cannot be determined from the given information

### Key Concepts

Algebra

Theory of Equations

Probability

## Check the Answer

AIME, 2015, Question 5

Geometry Revisited by Coxeter

## Try with Hints

First hint

Wednesday case - with restriction , select the pair on wednesday in $5 \choose 1$ ways

Tuesday case - four pair of socks out of which a pair on tuesday where a pair is not allowed where 4 pairs are left,the number of ways in which this can be done is $8 \choose 2$ - 4

Second Hint

Monday case - a total of 6 socks and a pair not picked $6 \choose 2$ -2

Final Step

by multiplication and principle of combinatorics $\frac{(5)({5\choose 2} -4)({6 \choose 2}-2)}{{10 \choose 2}{8 \choose 2}{6 \choose 2}}$=$\frac{26}{315}$. That is 341.

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Cheenta is a knowledge partner of Aditya Birla Education Academy