Tutors Answer Your Questions about Permutations (FREE)
Question 1167300: a) How many seven-digit telephone numbers have one digit which is a multiple of 4 and six digits which are
not a multiple of 4?
b) How many seven-digit telephone numbers have three digits which are a multiple of 4 and four digits which
are not a multiple of 4?
c) Continuing the pattern, and adding the disjoint possibilities, answer the broader question: How many
seven-digit telephone numbers have exactly an odd number of digits which are a multiple of 4?
Click here to see answer by ikleyn(52776)   |
Question 1175169: For the upcoming world-cup, the Indian Cricket Selection Committee has to come up with a possible batting order for their players. Instead of using the traditional approach they have decided to use computer algorithms to come up with all the possible batting orders and then decide from that. The
algorithm however requires the possible batting positions for each player.
The algorithm takes a list of 11 players. Each player can have more than one position they can bat at. Your job for now is to help the selection committee calculate the total number of unique batting charts such that every player gets exactly one batting position from their list of positions and no two players are given the same batting position in one batting chart.
Player / < position 1> / < position 2> / < position 3>….
Ex:
P1 / 1 / 2 / 3 / 4
P2 / 1 / 5 / 9 / 2 / 6 / 7 / 8
P3 / 1 / 2 / 7 / 10 / 3
P4 / 1 / 9 / 2 / 6 / 7 / 10 / 3 / 4
P5 / 5 / 9 / 2 / 8 / 3 / 4
P6 / 1 / 5 / 3 / 6
P7 / 6 / 7 / 4
P8 / 1 / 9 / 2 / 4
P9 / 9 / 6 / 11 / 3 / 4
P10 / 1 / 5 / 9 / 7 / 8 / 4
P11 / 6 / 11 / 7 / 10
The total number of allocations possible is: 4646.
How to arrive at this solution?
Click here to see answer by ikleyn(52776) |
Question 1210232: Recall that a partition of a positive integer n means a way of writing n as the sum of some positive integers, where the order of the parts does not matter. For example, there are five partitions of 4:
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
How many partitions of 17 are there that have at least three parts, such that the largest, second-largest, third-largest, and fourth-largest parts are respectively greater than or equal to 4, 3, 2, and 1?
The partition 17 = 7 + 4 + 3 + 2 + 1 is one such partition.)
Click here to see answer by ikleyn(52776) |
Question 1210232: Recall that a partition of a positive integer n means a way of writing n as the sum of some positive integers, where the order of the parts does not matter. For example, there are five partitions of 4:
4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1
How many partitions of 17 are there that have at least three parts, such that the largest, second-largest, third-largest, and fourth-largest parts are respectively greater than or equal to 4, 3, 2, and 1?
The partition 17 = 7 + 4 + 3 + 2 + 1 is one such partition.)
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 ikleyn(52776) |
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) |
Question 1210234: Vera has 20 white socks, 21 black socks, 22 brown socks, 23 blue socks, 24 red socks, and 25 green socks. How many socks (at a minimum) must she pull out of her sock drawer to ensure at least six matching pairs of different colors?
Click here to see answer by ikleyn(52776) |
Question 1210234: Vera has 20 white socks, 21 black socks, 22 brown socks, 23 blue socks, 24 red socks, and 25 green socks. How many socks (at a minimum) must she pull out of her sock drawer to ensure at least six matching pairs of different colors?
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 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 EPM(3) |
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 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 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 greenestamps(13198)  |
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 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 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 greenestamps(13198) |
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 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 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) |
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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
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