1. There are 4 red blocks and 2 blue blocks. How many patterns can you make by placing them in a line?

    $\binom{ 8}{ 1} = 8$ $\binom{ 4}{ 2} = 6$ $\binom{ 6}{ 4} = 15$ $\binom{ 6}{ 1} = 6$

  2. Two six-sided dice are thrown sequentially, and the face values that come up are recorded. Let $A$ be the event that the sum of two values is 5. Then find $P(A^c)$.

    $\displaystyle {{11}\over{12}}$ $\displaystyle {{31}\over{36}}$ $\displaystyle {{8}\over{9}}$ $\displaystyle {{13}\over{15}}$

  3. Suppose that $P(A) = {{3}\over{5}}$, $P(B) = {{7}\over{10}}$, and $P(A\cap B) = {{2}\over{5}}$. Find $P(A^c\cap B^c)$.

    $\displaystyle {{3}\over{5}}$ $\displaystyle {{1}\over{10}}$ $\displaystyle {{1}\over{2}}$ $\displaystyle {{9}\over{10}}$

  4. From a group of 7 men and 7 women, how many different committees consisting of 4 men and 5 women can be formed?

    $\binom{ 7}{ 4} \times \binom{ 7}{ 5} = 735$ $\binom{ 7}{ 4} + \binom{ 7}{ 5} = 56$ $\binom{ 14}{ 9} = 2002$ $\binom{ 7}{ 1} \times \binom{ 7}{ 1} = 49$

  5. A coin is tossed repeatedly until heads are observed 5 times. How many different outcomes (or patterns) are possible if the 5-th heads is observed at 8-th attempt of coin-tossing.

    $\binom{ 8}{ 5} = 56$ $\binom{ 7}{ 5} = 21$ $\binom{ 7}{ 4} = 35$ $\binom{ 8}{ 4} = 70$

  6. Suppose that $P(A) = {{1}\over{2}}$, $P(B) = {{2}\over{5}}$, and $P(A\cap B) = {{3}\over{10}}$. Find $P(A^c\cup B^c)$.

    $\displaystyle {{3}\over{10}}$ $\displaystyle {{2}\over{5}}$ $\displaystyle {{2}\over{5}}$ $\displaystyle {{7}\over{10}}$

  7. Find the coefficient of $\displaystyle x^5\,y^5$ in the expansion of $\displaystyle \left(y+x\right)^{10}$ .

    $\binom{ 25}{ 1} = 25$ $\binom{ 10}{ 5} = 252$ $\binom{ 5}{ 5} = 1$ $\binom{ 10}{ 1} = 10$

  8. Two six-sided dice are thrown sequentially, and the face values that come up are recorded. Let $A$ be the event that the sum of two values is 9. Then find $P(A)$.

    $\displaystyle {{1}\over{9}}$ $\displaystyle {{5}\over{36}}$ $\displaystyle {{1}\over{12}}$ $\displaystyle {{2}\over{15}}$

  9. Suppose that $P(A) = {{3}\over{10}}$, $P(B) = {{1}\over{2}}$, and $P(A\cap B) = {{1}\over{5}}$. Find $P(A\cap B^c)$.

    $\displaystyle {{9}\over{10}}$ $\displaystyle {{1}\over{10}}$ $\displaystyle {{1}\over{5}}$ $\displaystyle {{3}\over{10}}$

  10. A committee of 3 must be formed from a group of 15. How many different committees are possible?

    $\binom{ 18}{ 15} = 816$ $\binom{ 15}{ 3} = 455$ $\binom{ 18}{ 3} = 816$ $\binom{ 15}{ 1} = 15$


Department of Mathematics
Last modified: 2025-03-18