Using Permutations and Combinations in Probability - Statistics and Probability

1. From $10$ students, how many ways can you choose a president, vice president, and secretary?

   $P(10,3)=720$. Order matters since positions are distinct.

2. Compute the number of $4$-digit codes using digits $0$-$9$ with no repetition allowed.

   $P(10,4)=10\cdot 9\cdot 8\cdot 7=5040$. 10 choices, then 9, then 8, then 7 (no repeats).

3. A $5$-digit code uses digits $0$-$9$ with no repetition. How many codes are possible?

   $P(10,5)=30240$. $P(10,5) = \frac{10!}{5!} = 10 \times 9 \times 8 \times 7 \times 6 = 30240$.

4. State the formula for permutations of $n$ items taken $r$ at a time, $P(n,r)$.

   $P(n,r)=\frac{n!}{(n-r)!}$. Order matters, so divide $n!$ by $(n-r)!$ to remove unused items.

5. Two cards are drawn without replacement. What is $P(\text{both aces})$?

   $\frac{\binom{4}{2}}{\binom{52}{2}}$. Choose 2 aces from 4 divided by all ways to choose 2 from 52.

6. Compute $P(8,3)$, the number of ordered selections of $3$ from $8$.

   $P(8,3)=8\cdot 7\cdot 6=336$. First 3 factors of $8!$: $8\cdot 7\cdot 6$.

7. Find the probability that a $5$-card hand contains at least $1$ ace (from a $52$-card deck).

   $1-\frac{C(48,5)}{C(52,5)}$. 1 minus probability of no aces (complement rule).

8. State the multiplication rule for counting outcomes in a multi-stage process.

   Total outcomes $=a_1a_2\cdots a_k$. Each stage multiplies the number of choices at that stage.

9. State the formula for the combination of $n$ objects taken $r$ at a time, $C(n,r)$.

   $C(n,r)=\frac{n!}{r!(n-r)!}$. Divides by $r!$ to remove order from permutations.

10. Find the probability of getting exactly $2$ heads in $3$ fair coin flips.

    $\frac{C(3,2)}{2^3}=\frac{3}{8}$. Choose 2 heads positions from 3, divide by $2^3$ total outcomes.

11. Compute $C(8,3)$, the number of unordered selections of $3$ from $8$.

    $C(8,3)=56$. $\frac{8!}{3!5!}=\frac{336}{6}=56$.

12. State the classical probability formula using equally likely outcomes.

    $P(E)=\frac{\text{favorable}}{\text{total}}$. Assumes all outcomes are equally likely to occur.

13. What is the factorial definition of $n!$ for integer $n\ge 1$, and what is $0!$?

    $n!=n(n-1)\cdots 2\cdot 1$ and $0!=1$. Product of integers down to 1; special case $0!=1$.

14. Identify whether order matters for a permutation versus a combination.

    Permutation: order matters; combination: order does not. Key distinction: ABC ≠ BAC for permutations, but ABC = BAC for combinations.

15. State the formula for the number of distinct permutations of $n$ items with repeats $n_1,n_2,\dots,n_k$.

    $\frac{n!}{n_1!n_2!\cdots n_k!}$. Divides $n!$ by factorials of repeat counts to eliminate duplicates.

16. Compute the number of ways to arrange $6$ distinct books on a shelf.

    $6!=720$. All arrangements of 6 distinct items: $6!$.

17. From $10$ students, how many ways can you choose a $3$-person committee?

    $\binom{10}{3}=120$. Order doesn't matter for committee selection.

18. What is the fundamental counting principle for $k$ stages with $n_1,\dots,n_k$ choices?

    $n_1\cdot n_2\cdots n_k$. Multiply choices at each stage for total outcomes.

19. Compute the number of ways to select a president, vice president, and secretary from $10$ people.

    $P(10,3)=720$. Order matters for distinct positions: $10\cdot 9\cdot 8$.

20. How many distinct arrangements of the letters in $\text{LEVEL}$ are there?

    $\frac{5!}{2!2!}=30$. LEVEL has 5 letters with L and E each appearing twice.

21. What is the relationship between permutations and combinations: express $P(n,r)$ using $\binom{n}{r}$?

    $P(n,r)=\binom{n}{r}r!$. Each combination has $r!$ permutations when order is considered.

22. State the addition rule for any two events $A$ and $B$.

    $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. Adds probabilities but subtracts overlap to avoid double-counting.

23. A box has $6$ red and $4$ blue marbles. Two are drawn without replacement. What is $P(\text{one red and one blue})$?

    $\frac{\binom{6}{1}\binom{4}{1}}{\binom{10}{2}}=\frac{8}{15}$. Ways to pick 1 red and 1 blue divided by all ways to pick 2.

24. Compute $P(7,3)$.

    $210$. $P(7,3) = \frac{7!}{4!} = \frac{5040}{24} = 210$.

25. State the complement rule for probability.

    $P(E^c)=1-P(E)$. The probability of not-E equals 1 minus the probability of E.

26. State the formula for combinations of $n$ items taken $r$ at a time, $\binom{n}{r}$.

    $\binom{n}{r}=\frac{n!}{r!(n-r)!}$. Order doesn't matter, so divide by $r!$ to remove duplicate arrangements.

27. Compute the number of distinct permutations of the letters in $\text{MISSISSIPPI}$.

    $\frac{11!}{4!4!2!}=34650$. Divide by factorials of repeated letters: M(1), I(4), S(4), P(2).

28. State the probability rule for independent events $A$ and $B$.

    $P(A\cap B)=P(A)P(B)$. For independent events, multiply their individual probabilities.

29. Find the probability that a $5$-card hand contains exactly $2$ aces (from a $52$-card deck).

    $\frac{C(4,2)C(48,3)}{C(52,5)}$. Choose 2 aces and 3 non-aces, divide by all 5-card hands.

30. State the formula for the permutation of $n$ objects taken $r$ at a time, $P(n,r)$.

    $P(n,r)=\frac{n!}{(n-r)!}$. Divides $n!$ by $(n-r)!$ to remove the unused $(n-r)$ objects.

31. Compute the number of $4$-digit codes using digits $0$-$9$ with repetition allowed.

    $10^4=10000$. Each position has 10 choices independently.

32. Find the probability of drawing $2$ hearts in $2$ draws without replacement from a $52$-card deck.

    $\frac{C(13,2)}{C(52,2)}=\frac{1}{17}$. Choose 2 from 13 hearts over choose 2 from 52 cards.

33. What is the probability formula using equally likely outcomes and counting?

    $P(A)=\frac{\#A}{\#S}$. Favorable outcomes divided by total outcomes.

34. Compute $\binom{7}{3}$.

    $35$. $\binom{7}{3} = \frac{7!}{3!4!} = \frac{5040}{6 \times 24} = 35$.

35. Find the probability of drawing $2$ aces in $2$ draws without replacement from a $52$-card deck.

    $\frac{C(4,2)}{C(52,2)}=\frac{1}{221}$. Choose 2 from 4 aces over choose 2 from 52 cards.

36. Compute the number of ways to choose a committee of $3$ from $10$ people.

    $C(10,3)=120$. Order doesn't matter for committee selection.

37. A $5$-digit code uses digits $0$-$9$ with repetition allowed. How many codes are possible?

    $10^5$. Each of 5 positions can be any of 10 digits: $10 \times 10 \times 10 \times 10 \times 10$.

38. Find the probability a random $4$-digit code (digits $0$-$9$) has no repeated digits.

    $\frac{P(10,4)}{10^4}=\frac{5040}{10000}=0.504$. No-repeat codes divided by all possible codes.

39. A $5$-card hand is drawn from a $52$-card deck. What is $P(\text{all hearts})$?

    $\frac{\binom{13}{5}}{\binom{52}{5}}$. Choose 5 hearts from 13 divided by all ways to choose 5 from 52.

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