On June 27, 2015, about 0800 mountain daylight time, a Cessna TR182, N4751S, was substantially damaged when it impacted terrain west of the Natrona County International Airport (KCPR), Casper, Wyoming. Visual meteorological conditions prevailed at the time of the accident. The flight was conducted under the provisions of 14 Code of Federal Regulations Part 91 without a flight plan. The private pilot, pilot-rated passenger, and one additional passenger were seriously injured. The flight had just departed and was en route to Jackson Hole Airport, (KJAC), Jackson, Wyoming.The airplane was cleared for takeoff from runway 26. About 300 feet above ground level the engine began to run rough and lose power. The pilot and pilot-rated passenger attempted to restore engine power but were unsuccessful. The pilot-rated passenger "took control" of the airplane and attempted to return to the airport. The engine continued to run rough and lose power, and they were unable to return to a runway to land. The airplane impacted terrain beyond the airport perimeter fence, just off of the departure end of runway 26. The left wing was bent and buckled, the fuselage was crushed up and aft, and the firewall was buckled. A postaccident examination of the engine revealed that the sheath for the carburetor heat cable was partially separated. A black rubber adhesive was at the joint between the sheath and the clamp joint but was not securing the clamp to the joint. The carburetor heat at the carburetor was partially on and the position could not be changed when actuated by the control knob inside the cabin of the airplane. A blue liquid, consistent in smell with 100 low lead aviation fuel, was drained from the engine sump and left wing. The samples were clear of visible water or particle contamination. A quarter cup of a clear liquid was drained from the right wing. No other fuel or liquid was recovered from the right wing. The fuel samples were sent to a laboratory for further examination. The analysis of the chemical composition of the fuel samples was consistent with previously measured aviation gasolines. An examination of the airframe, engine, and related systems revealed no further anomalies. Temperature at the time of the accident was 18 degrees Celsius (C) (64 degrees Fahrenheit (F)) and a dewpoint of 10 degrees C (50 degrees F). A review of the carburetor icing probability chart, located in the Federal Aviation Administration's Special Airworthiness Information Bulletin CE-09-35, revealed that conditions at the time of the accident were conducive to the accumulation of serious carburetor icing at glide power.