7. Prove by mathematical induction that for each $n \ge 3$, if $A = \{1, \dots, n\}$ then the number of sets of the form $\{i, j, k\}$, where $1 \le i < j < k \le n$, is $\frac{n(n-1)(n-2)}{6}$. In this problem, you are allowed to use the fact that A has $\frac{n(n-1)}{2}$ subsets of the form $\{i, j\}$ with $1 \le i < j \le n$.
Added by Derrick G.
Close
Step 1
In this case, A = {1, 2, 3, 4}. We need to show that there exists a pair {i, j} such that 1 < i < j < 4. We can choose i = 2 and j = 3. This pair satisfies the conditions 1 < i < j < 4, so the statement holds true for n = 4. Show more…
Show all steps
Your feedback will help us improve your experience
Kevin Corkran-Itagaki and 58 other AP CS educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Show that n! > 3 n for n ≥ 7 (using mathematical Induction )?
Sri K.
Prove the following by using the principle of mathematical induction for all $n \in \mathbf{N}$. $$ (2 n+7)<(n+3)^{2} $$
Principle of Mathematical Induction
Introduction
(17) Prove by induction. For all integers n ≥ 7, n! ≥ 3^n + 2^n . (Hint: You will need 7! = 5, 040, 3^7 = 2, 187 and 2^7 = 128.)
Supreeta N.
Recommended Textbooks
Computer Science and Information Technology
Introduction to Programming Using Python
Computer Science - An Overview
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD