By Smelyakov S.V., Stoyan Y.G.

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To understand the solution, we first recall a special type of numbers called prime numbers and state an important result relating natural numbers and prime numbers. A natural number p is said to be prime (or called a prime) if p > 2 and the only divisors of p are 1 and p. All prime numbers less than 100 are shown below: 2,3,5,7,11,13,17,19,23,29,31,37,41, 43,47,53,59,61,67,71,73,79,83,89,97. The primes are often referred to as building blocks of numbers because every natural number can always be expressed uniquely as a product of some primes.

10 A team of 6 people is to be chosen from a list of 10 candidates. Find in how many ways this can be done (i) if the order of the people in the team does not matter; (ii) if the team consists of 6 people in a definite order; (iii) if the team consists of a first pair, a second pair and a third pair but order within each pair does not matter. /rr. 11 Find how many three figure numbers, lying between 100 and 999 inclusive, have two and only two consecutive figures identical. 12 33 Find t h e number of ways in which 10 persons can be divided into (i) two groups consisting of 7 and 3 persons; (ii) three groups consisting of 4, 3 and 2 persons with 1 person rejected.

Obviously, this correspondence establishes a bijection between the set of ways of distributing the balls and the set of 9-digit binary sequences with two l's. Thus, by (BP), the number of ways of distributing the seven identical balls into three distinct boxes is ( 2 ). 6). In the distribution problem discussed above, some boxes may be vacant at the end. Supposing no box is allowed to be vacant, how many ways are there to distribute the seven identical balls into three distinct boxes? To meet the requirement that no box is vacant, we first put a ball in each box and this is counted as one way because the balls are identical.

### A basis of equivalence classes of paths in optimization problems by Smelyakov S.V., Stoyan Y.G.

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