Table Method with Six Tiles (Grades 7-8)

(Listing all of the possible 3-number combinations that can be drawn; highlighting the outcomes of 7.)

There are 216 possible 3-number (3-draw number) combinations.  Of these 216 possible three-draws, 27 end up with an outcome of 7 (i.e., the two highest numbers add to 7).  The possibility of obtaining an outcome of 7 is 27/216 or 1/8.

 111 112 113 114 115 116 211 212 213 214 215 216 311 312 313 314 315 316 411 412 413 414 415 416 511 512 513 514 515 516 611 612 613 614 615 616 121 122 123 124 125 126 221 222 223 224 225 226 321 322 323 324 325 326 421 422 423 424 425 426 521 522 523 524 525 526 621 622 623 624 625 626 131 132 133 134 135 136 231 232 233 234 235 236 331 332 333 334 335 336 431 432 433 434 435 436 531 532 533 534 535 536 631 632 633 634 635 636 141 142 143 144 145 146 241 242 243 244 245 246 341 342 343 344 345 346 441 442 443 444 445 446 541 542 543 544 545 546 641 642 643 644 645 646 151 152 153 154 155 156 251 252 253 254 255 256 351 352 353 354 355 356 451 452 453 454 455 456 551 552 553 554 555 556 651 652 653 654 655 656 161 162 163 164 165 166 261 262 263 264 265 266 361 362 363 364 365 366 461 462 463 464 465 466 561 562 563 564 565 566 661 662 663 664 665 666

(Determining the possible 3-number combinations by calculation;

determining the number of outcomes of 7 by reasoning and calculating.)

Total number of 3-number combinations possible:

6 x 6 x 6  216

This is because there are 6 possible numbers for the first tile, each of which can be followed by 6 possible numbers for the second tile.  These 36 pairs can each be followed by 6 possible numbers for the third tile.  This makes a total of 36 x 6 = 216 combinations total.

Total number of outcomes of 7 possible:

The possible combinations of 3 numbers that could be combined to get a 7 from the highest two draws are:

611  (No other higher numbers would work with a 6; only 6 and 1 equal 7.0

522  (No other higher numbers would work with the 5; only 5 and 2 followed by 2 or 1.)

521  (See above.)

433  (No other higher numbers would work with the 4; only 4 and 3 followed by 3, 2, o4 1.)

432  (See above.)

431  (See above.)

Starting with a 3, 2, or 1 would result in the same groups of 3 numbers already listed, only in different order.

These 6 trios of numbers can, now, be accomplished in different orders, as follows:

611:  611, 161, 116                                                                             (3 ways)

522   522, 252, 225                                                                             (3 ways)

521:  521, 512, 251, 215, 152, 125                                                    (6 ways)

433:  433, 343, 334                                                                             (3 ways)

432:  432, 423, 342, 324, 243, 234                                                    (6 ways)

431:  431, 413, 341, 314, 134, 143                                                + (6 ways)

27 ways

Out of the 216 possible 3-number combinations (tiles draws), 27 will result in an outcome of 7.  27/216  or 1/8  is the probability of obtaining an outcome of 7.