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Question 1210186: Find the number of positive integers that are divisors of at least one of 6^{6}, 10^{10}, 15^{15}, and 30^{30}.

Click here to see answer by AnlytcPhil(1806)

Question 1210199: Find the number of sequences (a_1, a_2, a_3, \dots, a_8) such that:
* a_i \in \{1, 2, 3, 4, 5, 6, 7, 8\} for all 1 \le i \le 8.
* Every number1, 2, 3, 4, 5, 6, 7, 8 appears at least once in the sequence.
Click here to see answer by ikleyn(52779)

Question 1210200: A permutation of the numbers (1,2,3,\dots,n) is a rearrangement of the numbers in which each number appears exactly once. For example, (2,5,1,4,3) is a permutation of (1,2,3,4,5).

Find the number of permutations (x_1, x_2, \dots, x_8) on (1, 2, 3, \dots, 8), such that (x_i, x_{i + 1}), 1 \le i \le 7, is never equal to (1,2), (3,4), (5,6), or (7,8). (However, (x_i, x_{i + 1}) can be equal to (2,1), (4,3), (6,5), or (8,7).)
Click here to see answer by CPhill(1959)

Question 1210198: A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits (spades, hearts, diamonds, and clubs), such that there is exactly one card for any given rank and suit.

You are dealt a hand of 13 cards. Find the probability that your hand has a void. (Your hand has a void if it does not contain any cards of a particular suit.)

Once you've computed the answer in terms of binomial coefficients, use a calculator or computer to determine the answer to the nearest tenth of a percent, and enter that as your answer.
Click here to see answer by ikleyn(52779)

Question 1210202: How many subsets of \{0, 1, \dots, 9\} have the property that there are at least two elements and the sum of the two largest elements is 4?
Click here to see answer by CPhill(1959)

Question 1210203: We choose a positive divisor of 20^{20} at random (with all divisors equally likely to be chosen). What is the probability that we chose a multiple of 5?
Click here to see answer by CPhill(1959)

Question 1210204: Find the number of ways of arranging one A, two Bs, three Cs, and four Ds, so that no two Bs are next to each other, no two Cs are next to each other, and no two Ds are next to each other.
Click here to see answer by CPhill(1959)

Question 1210205: How many 7-digit sequences have a digit that appears at least 6 times?

(For example, 3339333 and 0200000 are two such sequences. A sequence is allowed to begin with 0.)

Click here to see answer by CPhill(1959)

Question 1165576: The "Pick 3" at horse racetracks requires that a person select the winning horse for three consecutive races. If the first race has ten entries, the second race seven entries, and the third race eleven entries, how many different possible tickets might be purchased?
Click here to see answer by ikleyn(52779)

Question 1210210: How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?
Click here to see answer by CPhill(1959)

Question 1210210: How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?
Click here to see answer by ikleyn(52779)

Question 1210211: How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?

Click here to see answer by CPhill(1959)

Question 1210212: How many positive integers are there whose digits strictly decrease from left to right, and have at most one even digit, and the sum of the digits is 6?
Click here to see answer by CPhill(1959)

Question 1210216: A bag contains red and blue tiles. Each tile has a number from the set \{-1, 0, 1\} written on it. I want to arrange 7 of these tiles in a row, so that the numbers on any three consecutive tiles sum to 3. In how many ways can this be done, assuming that there are an unlimited number of tiles for any color and number combination?

Click here to see answer by CPhill(1959)

Question 1210217: How many ordered pairs of positive integers (m,n) satisfy \text{lcm}[m,n] = 360 and \gcd(m,n) = 360?
Click here to see answer by CPhill(1959)

Question 1210218: Find the number of ways of filling in the squares of a 3 \times 3 grid so that:
* Each square contains a 0 or a 1.
* The sum of the numbers in each row and each column is at most 1.

An example is shown below.
Click here to see answer by CPhill(1959)

Question 1210219: Find the number of paths from A to B in the grid below, so that
* Each step is down or to the right.
* The path cannot pass through any point more than once.

An example path is shown.

The grid is 3 by 3, with A at the upper-left, and B at the lower-right.
Click here to see answer by CPhill(1959)

Question 1210183: In how many ways can we seat 3 pairs of siblings in a row of 10 chairs, so that nobody sits next to their sibling? (Two chairs will be left empty, of course.)

Click here to see answer by ikleyn(52779)

Question 1168178: A rumor is spread randomly among a group of 10 people by successively having one person call someone, who calls someone, and so on. A person can pass the rumor on to anyone except the person who just called. (a) By how many different paths can a rumor travel through the group in three calls? In "n" calls? (b) what is the probability that if A starts the rumor, A receives the third calls? (c) What is the probability that if A does not start the rumor, A receives the third call?
Click here to see answer by CPhill(1959)

Question 1210221: Find the number of paths from A to B in the grid below, where each step is down or to the right.

The grid is 3 by 3, with A in the upper-left and B in the lower-right.
Click here to see answer by CPhill(1959)

Question 1210222: Find the number of arithmetic sequences such that:
* The arithmetic sequence contains three terms
* All the terms are integers in \{0, 1, 2, \dots, 10\}
* The sum of the terms is 9.
Click here to see answer by CPhill(1959)

Question 1210224: Find the number of sequences containing three terms, such that
* The second term is equal to the sum of the first term plus one.
* The third term is equal to twice the second term.
* Each term is an integer in \{0, 1, 2, \dots, 100\}.

Click here to see answer by CPhill(1959)

Question 1210229: Find the number of subsets of
S = \{1, 3, 8, 17, 30, 36, 47, 58\},
so that the sum of the elements in the subset is less than 20. (Note that for the empty subset, we take the sum of the elements as 0.)

Click here to see answer by CPhill(1959)

Question 1210228: Find the number of subsets of
S = \{1, 3, 8, 17, 30, 36, 47, 58\},
so that the sum of the elements in the subset is a multiple of 5. (Note that for the empty subset, we take the sum of the elements as 0.)

Click here to see answer by CPhill(1959)

Question 1210230: Consider the set
S = {1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 18, 23, 24, ..., 12345678},
which consists of all positive integers whose digits strictly increase from left to right, and the digits are from 1 to 8. This set is finite. What is the sum of the elements of the set?
Click here to see answer by CPhill(1959)

Question 1210231: Find the number of ways filling in a 4 \times 4 grid, such that
* Each cell contains a 0 or a 1.
* The sum of the numbers in each row and each column is at least 2.
An example is shown below.

0110
1010
0011
1111
Click here to see answer by CPhill(1959)

Older solutions: 1..45, 46..90, 91..135, 136..180, 181..225, 226..270, 271..315, 316..360, 361..405, 406..450, 451..495, 496..540, 541..585, 586..630, 631..675, 676..720, 721..765, 766..810, 811..855, 856..900, 901..945, 946..990, 991..1035, 1036..1080, 1081..1125, 1126..1170, 1171..1215, 1216..1260, 1261..1305, 1306..1350, 1351..1395, 1396..1440, 1441..1485, 1486..1530, 1531..1575, 1576..1620, 1621..1665, 1666..1710, 1711..1755, 1756..1800, 1801..1845, 1846..1890, 1891..1935, 1936..1980, 1981..2025, 2026..2070, 2071..2115, 2116..2160, 2161..2205, 2206..2250, 2251..2295, 2296..2340, 2341..2385, 2386..2430, 2431..2475, 2476..2520, 2521..2565, 2566..2610, 2611..2655, 2656..2700, 2701..2745, 2746..2790, 2791..2835, 2836..2880, 2881..2925, 2926..2970, 2971..3015, 3016..3060, 3061..3105, 3106..3150, 3151..3195, 3196..3240, 3241..3285, 3286..3330, 3331..3375, 3376..3420, 3421..3465, 3466..3510, 3511..3555, 3556..3600, 3601..3645, 3646..3690, 3691..3735, 3736..3780, 3781..3825, 3826..3870, 3871..3915, 3916..3960, 3961..4005, 4006..4050, 4051..4095, 4096..4140, 4141..4185, 4186..4230, 4231..4275, 4276..4320, 4321..4365, 4366..4410, 4411..4455, 4456..4500, 4501..4545, 4546..4590, 4591..4635, 4636..4680, 4681..4725, 4726..4770, 4771..4815, 4816..4860, 4861..4905, 4906..4950, 4951..4995, 4996..5040, 5041..5085, 5086..5130, 5131..5175, 5176..5220, 5221..5265, 5266..5310, 5311..5355, 5356..5400, 5401..5445, 5446..5490, 5491..5535, 5536..5580, 5581..5625, 5626..5670, 5671..5715, 5716..5760, 5761..5805, 5806..5850, 5851..5895, 5896..5940, 5941..5985, 5986..6030, 6031..6075, 6076..6120, 6121..6165, 6166..6210, 6211..6255, 6256..6300, 6301..6345, 6346..6390, 6391..6435, 6436..6480, 6481..6525, 6526..6570, 6571..6615, 6616..6660, 6661..6705, 6706..6750, 6751..6795, 6796..6840, 6841..6885, 6886..6930, 6931..6975, 6976..7020, 7021..7065, 7066..7110, 7111..7155, 7156..7200, 7201..7245, 7246..7290, 7291..7335, 7336..7380, 7381..7425, 7426..7470, 7471..7515, 7516..7560, 7561..7605, 7606..7650, 7651..7695, 7696..7740, 7741..7785, 7786..7830, 7831..7875, 7876..7920, 7921..7965, 7966..8010, 8011..8055, 8056..8100, 8101..8145, 8146..8190, 8191..8235, 8236..8280, 8281..8325, 8326..8370, 8371..8415, 8416..8460, 8461..8505, 8506..8550, 8551..8595, 8596..8640, 8641..8685, 8686..8730, 8731..8775, 8776..8820, 8821..8865, 8866..8910, 8911..8955, 8956..9000, 9001..9045, 9046..9090, 9091..9135, 9136..9180, 9181..9225, 9226..9270, 9271..9315, 9316..9360, 9361..9405, 9406..9450, 9451..9495, 9496..9540, 9541..9585, 9586..9630, 9631..9675, 9676..9720, 9721..9765, 9766..9810, 9811..9855, 9856..9900, 9901..9945, 9946..9990, 9991..10035