GROUP #1: Fundamental Principle of Counting 1 How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) repetition of the digits is allowed. (ii) repetition of the digits is not allowed.

using Permutations & Combinations

p6

plz solve this question in detail

5) Using mathematical induction, show that for all integers P(n): n > 8, n can be written as a sum of 3's or a sum of 5's or a sum of a combination of 3's and 5's.

Find the sum of the first five natural even numbers using sigma notation.

​(a) If repetition of digits is not​ allowed, how many four​-digit numbers can be formed using the digits​ 1, 2, 3, 4, and 5​?   ​(b) How many of the numbers in part​ (a) begin with either of the digits 2, 3, or 4​?   ​(c) How many of the numbers in part​ (a) are even​?

numbers of four digits can be formed with the digits 0, 1, 2, 3 (repetition of digits is How many not allowed? (a) 18 (b) 24

Fibonacci's Rabbits Leonardo Fibonacci da Pisa (c. 1170-1250), an outstanding European mathematician of the middle ages stated an interesting problem in his main work Liber Abaci (1202) where he wanted to know how many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair brings forth a new pair that becomes productive from the second month on and assuming no pairs die. Therefore, starting with F1 = 1 newly born pair in the first month, we can find the number Fk of pairs in the kth month. Using some knowledge of calculus, acquired when studying sequences, we can determine the number of pairs of rabbits in the kth month by setting up the following equation: Fk = (Number of pairs alive the preceding month) + (Number of newly born pairs for the kth month) Since the rabbits do not produce offsprings during the first month of their lives, we can see that the number of newly born pairs for the kth month is the number Fk - 2 of pairs alive…

How many ways can n different flags be flown on m flag poles in a row if any number of flags may be on the same pole, and order from the top to bottom matters?