C960 Discrete Mathematics II

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Free C960 Discrete Mathematics II Questions

How many 5-card poker hands contain exactly three of a kind (but not a full house)?

54,912
3744
123,552
2,598,960

Explanation

Exactly three of a kind means the hand contains three cards of one rank and two other cards of different ranks (and those two must not form a pair, otherwise it would be a full house). Count by choices: pick the rank for the triple $(\begin{matrix} 13 \\ 1 \end{matrix}) = 13$ , pick which 3 of the 4 suits for that rank $((\begin{matrix} 4 \\ 3 \end{matrix}) = 4)$ , choose 2 distinct ranks from the remaining 12 ranks for the other cards $((\begin{matrix} 12 \\ 2 \end{matrix}) = 66)$ , and for each of those ranks choose 1 of the 4 suits $4^{2} = 16$ . Multiply: 13×4×66×16=54,912. (For reference, the total number of 5-card hands is 2,598,960, which shows this count is a fraction of all possible hands.)

What is the space complexity of depth-first search (DFS) on a graph with $V$ vertices and $E$ edges?

O(V + E)
O(V²)
O(V)
O(E)

Explanation

Depth-first search (DFS) uses a stack to keep track of vertices during traversal, which can be implemented either recursively (call stack) or iteratively (explicit stack). In addition to the stack, DFS may store the adjacency list or adjacency matrix of the graph. For a graph represented as an adjacency list, storing all edges takes $O (V + E)$ space, and the stack can grow up to $O (V)$ in the worst case. Combining these, the overall space complexity of DFS on a graph with $v$ vertices and $E$ edges is $O (V + E)$

What is the expected number of coin flips needed to get the sequence HTH?

Explanation

How many edges does a complete binary tree with 31 nodes have?

Explanation

In any tree, the number of edges is always one less than the number of nodes because a tree is a connected acyclic graph. Therefore, for a tree with $n$ nodes, the number of edges is $n - 1$ . Here, the tree has 31 nodes, so the number of edges is:

What is the probability that exactly 3 out of 5 independent fair coin flips show heads before the first tail appears in a sequence?

1/8
1/16
1/4
1/32

Explanation

The event "exactly 3 heads before the first tail" means the first tail occurs on the 4th flip, and the first 3 flips are all heads. For fair coins, the probability of a head or tail on any flip is $\frac{1}{2}$ . The probability of three heads followed by a tail is:

P(HHHT)= ${(\frac{1}{2})}^{3} . (\frac{1}{2}) = {(\frac{1}{2})}^{4} = \frac{1}{16}$

So the probability is $\frac{1}{16}$ .

In a group of 30 people, what is the minimum number that guarantees at least 4 people share the same birth month?

Explanation

This is an application of the Pigeonhole Principle. There are 12 months (pigeonholes) and 30 people (pigeons). To guarantee at least 4 people in the same month, consider the worst-case scenario where no month has 4 people: each month could have at most 3 people.

12 months×3 people per month=36

Wait, we need the minimum number that guarantees 4 people in a month. In the worst case, if each month has at most 3 people, then 12 × 3 = 36 people could avoid having 4 in a month. But we only have 30 people, which is less than 36.

To guarantee 4 people in a month, the minimum number of people needed is 3 × 12 + 1 = 37.

Since we have only 30 people, we cannot guarantee 4 in a month.

If the question is asking more generally: the formula is:

Minimum number to guarantee k in a box=(k−1)⋅n+1

Here, k=4, n=12:

(4−1)⋅12+1=37

What is the generating function for the number of ways to climb n stairs taking 1, 2, or 3 steps at a time?

$\frac{1}{1 - x - x^{2} - x^{3}}$
$\frac{1}{1 - x}$
$\frac{1}{{(1 - x)}^{3}}$
$\frac{1}{1 - 3 x}$

Explanation

What is the output of this greedy algorithm for the coin change problem (coins: 1, 5, 10, 25) when making 67 cents?

2 quarters, 1 dime, 1 nickel, 2 pennies
2 quarters, 1 dime, 1 nickel, 3 pennies
2 quarters, 1 dime, 1 nickel, 7 pennies
2 quarters, 1 dime, 1 nickel, 1 penny

Explanation

In the greedy algorithm for the coin change problem, we always choose the largest coin denomination that does not exceed the remaining amount. Starting with 67 cents:

Take 2 quarters (25 × 2 = 50) → remaining 17 cents.

Take 1 dime (10) → remaining 7 cents.

Take 1 nickel (5) → remaining 2 cents.

Take 2 pennies (1 × 2 = 2) → remaining 0 cents.

So the algorithm selects: 2 quarters, 1 dime, 1 nickel, and 2 pennies. This greedy approach gives the optimal result for standard U.S. coin denominations.

What is the value of the partition function p(8) (number of ways to write 8 as a sum of positive integers, disregarding order)?

Explanation

The partition function p(n)p(n)p(n) counts the number of ways to write nnn as a sum of positive integers, ignoring the order of the summands. For n=8n = 8n=8, the partitions are:

1.8

2.7 + 1

3.6 + 2

4.6 + 1 + 1

5.5 + 3

6.5 + 2 + 1

7.5 + 1 + 1 + 1

8.4 + 4

9.4 + 3 + 1

10.4 + 2 + 2

11.4 + 2 + 1 + 1

12.4 + 1 + 1 + 1 + 1

13.3 + 3 + 2

14.3 + 3 + 1 + 1

15.3 + 2 + 2 + 1

16.3 + 2 + 1 + 1 + 1

17.3 + 1 + 1 + 1 + 1 + 1

18.2 + 2 + 2 + 2

19.2 + 2 + 2 + 1 + 1

20.2 + 2 + 1 + 1 + 1 + 1

21.2 + 1 + 1 + 1 + 1 + 1 + 1

22.1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

Counting carefully, there are 22 distinct partitions of 8.

10.

What is the probability that a random permutation of {1,2,…,10} has exactly 3 cycles?

210/10!
9330/10!
3410/10!
1023/10!

Explanation

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