Question 949376: Find how many different odd 4-digit numbers less than 4000 can be made from the digits
1, 2, 3, 4, 5, 6, 7 if no digit may be repeated.
Answer by Edwin McCravy(20055)   (Show Source):
You can put this solution on YOUR website! Find how many different odd 4-digit numbers less than 4000 can be made from the digits
1, 2, 3, 4, 5, 6, 7 if no digit may be repeated.
Always take care of the most restricted things first. We
have two restrictions. The first digit must be from the set
{1,2,3} and the last digit must be from the set {1,3,5,7}
Since 1 and 3 are in both sets, we have two cases.
Case 1. Numbers with the first digit 1 or 3.
Case 2. Numbers with the first digit 2.
Case 1.
Choose the first digit 2 ways, (as 1 or 3)
Choose the fourth digit 3 ways, from either {3,5,7} or {1,5,7}
Choose the second digit any of the 5 remaining letters.
Choose the third digit any of the 4 remaining letters.
That's  ways for Case 1.
Case 2.
Choose the first digit 1 way, as 2.
Choose the fourth digit 4 ways, from {1,3,5,7}
Choose the second digit any of the 5 remaining letters.
Choose the third digit any of the 4 remaining letters.
That's  ways for Case 2.
Total: 120+80 = 200.
FYI, here is a computer listing of all 200 numbers,
20 rows of 10 each:
1235 1237 1243 1245 1247 1253 1257 1263 1265 1267
1273 1275 1325 1327 1345 1347 1357 1365 1367 1375
1423 1425 1427 1435 1437 1453 1457 1463 1465 1467
1473 1475 1523 1527 1537 1543 1547 1563 1567 1573
1623 1625 1627 1635 1637 1643 1645 1647 1653 1657
1673 1675 1723 1725 1735 1743 1745 1753 1763 1765
2135 2137 2143 2145 2147 2153 2157 2163 2165 2167
2173 2175 2315 2317 2341 2345 2347 2351 2357 2361
2365 2367 2371 2375 2413 2415 2417 2431 2435 2437
2451 2453 2457 2461 2463 2465 2467 2471 2473 2475
2513 2517 2531 2537 2541 2543 2547 2561 2563 2567
2571 2573 2613 2615 2617 2631 2635 2637 2641 2643
2645 2647 2651 2653 2657 2671 2673 2675 2713 2715
2731 2735 2741 2743 2745 2751 2753 2761 2763 2765
3125 3127 3145 3147 3157 3165 3167 3175 3215 3217
3241 3245 3247 3251 3257 3261 3265 3267 3271 3275
3415 3417 3421 3425 3427 3451 3457 3461 3465 3467
3471 3475 3517 3521 3527 3541 3547 3561 3567 3571
3615 3617 3621 3625 3627 3641 3645 3647 3651 3657
3671 3675 3715 3721 3725 3741 3745 3751 3761 3765
Edwin
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