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94 problems match “counting”. Clear

Ten numbered coins sum at least 45 ●●●○○ 50 ptsYou have 10 coins. Coin $i$ shows the number $i$ on one side (for $i = 1, \dots, 10$ ) and $0$ on the other side. You toss all the coins and add up the values that face up. Find the probability that the sum is greater than or equal to $45$ . Give your answer as a decimal.
Counting nine-term sequences with ratio constraints ●●●○○ 50 ptsThe sequence $\{a_n\}$ has nine terms, $a_1=a_9=1$ , and for each $i\in\{1,2,\ldots,8\}$ we have $\frac{a_{i+1}}{a_i}\in\left\{2,1,-\frac{1}{2}\right\}$ . How many such sequences are there?
Position of a fraction in a sequence ●●●○○ 50 ptsThe positive rational numbers are arranged in the order $\frac{1}{1}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{2}{2}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2}, \frac{2}{3}, \frac{1}{4}, \cdots$ . Find the position number of the number $\frac{1989}{1949}$ in this sequence.
Product of the two final numbers ●●●○○ 50 ptsThere are a total of 10040 numbers written on a blackboard: 2006 ones, 2007 twos, 2008 threes, 2009 fours and 2010 fives. Each operation erases four different numbers, one of each of four types, and writes one number of the fifth type (for example, erase one 1, one 2, one 3 and one 4, then write one 5; or erase one 2, one 3, one 4 and one 5, then write one 1). After a finite number of operations exactly two numbers remain on the blackboard. Find the product of these two numbers.
Children with siblings in the class ●●●○○ 50 ptsIn a class, exactly 18 fathers and exactly 24 mothers attended a parent-teacher meeting, and at least one parent of each student attended. Exactly 10 boys and exactly 8 girls had both parents present. For exactly 4 boys and exactly 3 girls only the mother was present, and for exactly 1 boy and exactly 1 girl only the father was present. There are no children in the class with step-parents or step-siblings. Determine the number of children in this class who have siblings in the same class.
Non-decreasing rolls of a die ●●●○○ 50 ptsA fair die is tossed 6 times. What is the probability that on each toss, the number shown is greater than or equal to the number on the previous toss? Give your answer as a decimal rounded to four decimal places.
Distinct values from parenthesised divisions ●●●○○ 50 ptsHow many different values can be obtained by placing parentheses in all possible ways in the expression $2:3:5:7:11:13:17:19:23:29$ , where $:$ denotes division?
Arranging letters of COMPUTE ●●●○○ 50 ptsHow many ways are there to arrange the letters of the word COMPUTE so that the vowels are in alphabetical order, or the consonants are in alphabetical order, or at least two vowels are next to each other?
Blue rectangles on a blackboard ●●●○○ 50 ptsBlue and red rectangles are drawn on a blackboard. Exactly 7 of the rectangles are squares. There are 3 more red rectangles than blue squares. There are 2 more red squares than blue rectangles. How many blue rectangles are there?
Counting alternating-parity increasing sequences ●●●○○ 50 ptsA finite strictly increasing sequence of integers is called alternating if its first term is odd, its second term is even, its third term is odd, and so on, alternating in parity. The empty sequence also counts as alternating. Let $A(n)$ be the number of alternating sequences whose terms are all taken from $\{1,2,\ldots,n\}$ , so $A(1)=2$ and $A(2)=3$ . Find $A(20)$ .