How many different 5-digit numbers can be formed with the digits 3,4,5,6, and 7, if each digit is used only once ?

120 numbers can be formed.

For the first digit you have five choices, for the second you have four choices (cuz you can't pick the same one again), for the third you have three choices, ..., for the last one there is only one choice.

5*4*3*2*1=120

For fun, here are all the 120 possible numbers:
1: 34567
2: 34576
3: 34657
4: 34675
5: 34756
6: 34765
7: 35467
8: 35476
9: 35647
10: 35674
11: 35746
12: 35764
13: 36457
14: 36475
15: 36547
16: 36574
17: 36745
18: 36754
19: 37456
20: 37465
21: 37546
22: 37564
23: 37645
24: 37654
25: 43567
26: 43576
27: 43657
28: 43675
29: 43756
30: 43765
31: 45367
32: 45376
33: 45637
34: 45673
35: 45736
36: 45763
37: 46357
38: 46375
39: 46537
40: 46573
41: 46735
42: 46753
43: 47356
44: 47365
45: 47536
46: 47563
47: 47635
48: 47653
49: 53467
50: 53476
51: 53647
52: 53674
53: 53746
54: 53764
55: 54367
56: 54376
57: 54637
58: 54673
59: 54736
60: 54763
61: 56347
62: 56374
63: 56437
64: 56473
65: 56734
66: 56743
67: 57346
68: 57364
69: 57436
70: 57463
71: 57634
72: 57643
73: 63457
74: 63475
75: 63547
76: 63574
77: 63745
78: 63754
79: 64357
80: 64375
81: 64537
82: 64573
83: 64735
84: 64753
85: 65347
86: 65374
87: 65437
88: 65473
89: 65734
90: 65743
91: 67345
92: 67354
93: 67435
94: 67453
95: 67534
96: 67543
97: 73456
98: 73465
99: 73546
100: 73564
101: 73645
102: 73654
103: 74356
104: 74365
105: 74536
106: 74563
107: 74635
108: 74653
109: 75346
110: 75364
111: 75436
112: 75463
113: 75634
114: 75643
115: 76345
116: 76354
117: 76435
118: 76453
119: 76534
120: 76543